# Automorphisms of real semisimple Lie algebras and their restricted root
systems

Ivan Solonenko  Department of Mathematics, King’s College London, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

We prove that every automorphism of the restricted root system of a real
semisimple Lie algebra – when defined properly – can be lifted to an
automorphism of that Lie algebra. In particular, this can be applied to
automorphisms of the Dynkin diagram of the restricted root system. We also
discuss some applications of this result to the theory of symmetric spaces of
noncompact type.

##  1\. Introduction

The correspondence between complex semisimple Lie algebras and reduced root
systems is a classical and pivotal result in Lie theory. Its discovery allowed
W. Killing and later E. Cartan to obtain the classification of complex simple
Lie algebras by reducing it to the level of root systems. It also proves
indispensable when one studies automorphisms of complex semisimple Lie
algebras, as those can be – in a sense – reduced to automorphisms of the
corresponding root system and its Dynkin diagram. One of the pillars of this
theory is the Isomorphism Theorem, which basically says that isomorphisms
between complex semisimple Lie algebras can be defined merely on the so-called
canonical generators – provided that the Cartan matrix is preserved. This
theorem can be used to lift any automorphism of the root system of
$\mathfrak{{g}}$ to an automorphism of $\mathfrak{{g}}$ itself. As a
consequence, one can show that the outer automorphism group of
$\mathfrak{{g}}$ is isomorphic to the automorphism group of its Dynkin
diagram.

Adjacent to the theory of complex semisimple Lie algebras is that of real
semisimple Lie algebras. Those were classified by E. Cartan, but the
classification is more intricate than in the complex case. Two standard
classification routes are by means of Vogan or Satake diagrams: both are
obtained by decorating the Dynkin diagram of the complexification of
$\mathfrak{{g}}$ in a certain way. There is another diagram that one can
associate to a real semisimple Lie algebra, namely the Dynkin diagram of its
restricted root system. The restricted root system of $\mathfrak{{g}}$ loses
all information about the compact ideals of $\mathfrak{{g}}$ and hence is not
suitable for classification. However, when treated properly, it can recover
the noncompact part of $\mathfrak{{g}}$, that is, the sum of all of its
noncompact simple ideals. The main disadvantage of this theory when compared
to the complex case is that there is no sensible analog of the Isomorphism
Theorem – there are no canonical generators to begin with. Yet, by using the
known classification of real semisimple Lie algebras, it is possible to prove
that every automorphism of the restricted root system of $\mathfrak{{g}}$ –
when defined properly – can be lifted to an automorphism of $\mathfrak{{g}}$.
This is the main purpose of the current paper. In the proof, we deliberately
eschew a case-by-case consideration. Instead, after reducing the problem to
simple real Lie algebras, we divide those into three different blocks –
depending on the nature of $\mathfrak{{g}}$ as a real form of its
complexification. This result appears to be somewhat folklore, as several
authors state it without giving any reference (see, e.g., [2, p. 11] or [9, p.
111]).

We also give a reformulation of this result for symmetric spaces of noncompact
type, as those are intimately related to noncompact semisimple Lie algebras.

This paper is organized as follows. In Section 2 we review some aspects of the
theory of root systems and their relation to complex semisimple Lie algebras.
We focus primarily on root system isomorphisms and the closely related notion
of Dynkin diagram isomorphisms. In Section 3 we look at real semisimple Lie
algebras and their restricted root systems. We introduce the notion of a
weighted isomorphism and prove that every weighted root system isomorphism
can, in a certain sense, be lifted to a Lie algebra isomorphism. In Section 4
we look at the results of Section 3 through the lens of symmetric spaces of
noncompact type. We also establish some links between a symmetric space of
noncompact type and its isometry Lie algebra.

##  2\. Root system isomorphisms and complex semisimple Lie algebras

Practically everything discussed in this section can be found in [7, Ch. II]
or [10, §1,4]. We begin by reviewing some aspects of the theory of root
systems. Let $(V,\Updelta)$ be a root system. Here $V$ is a finite-dimensional
Euclidean real vector space and $\Updelta\subseteq V$ is the root system
itself. We are not assuming $\Updelta$ to be reduced or irreducible. First, we
recall the notion of isomorphism of root systems.

######  Definition 1.

Let $(V^{\prime},\Updelta^{\prime})$ be another root system. A linear
isomorphism $f\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
is called a (root system) isomorphism between $\boldsymbol{(V,\Updelta)}$ and
$\boldsymbol{(V^{\prime},\Updelta^{\prime})}$ (or between
$\boldsymbol{\Updelta}$ and $\boldsymbol{\Updelta^{\prime}}$, for brevity) if
the following two conditions are satisfied:

  1. (i)

$f(\Updelta)=\Updelta^{\prime}$;

  2. (ii)

$f$ preserves the root integers, i.e.
$n_{f(\upalpha)f(\upbeta)}=n_{\upalpha\upbeta}$ for all
$\upalpha,\upbeta\in\Updelta$ (here $n_{\upalpha\upbeta}=\frac{2\langle\hskip
0.70004pt{\upalpha}\hskip 0.70004pt|\hskip 0.70004pt{\upbeta}\hskip
0.70004pt\rangle\hskip 0.14003pt}{||\upbeta||^{2}}$);

If $V^{\prime}=V$ and $\Updelta^{\prime}=\Updelta$, we call $f$ an
automorphism of $\boldsymbol{(V,\Updelta)}$ (or of $\boldsymbol{\Updelta}$,
for brevity). The (finite) group of all automorphisms of $\Updelta$ is denoted
by $\boldsymbol{\mathrm{Aut}\hskip
0.59998pt(\Updelta)}\subseteq\mathrm{GL}(V)$.

Note that, assuming (i), condition (ii) is automatically satisfied if $f$ is
conformal (i.e., a scalar multiple of an isometry). Although a root system
isomorphism does not have to be conformal in general, we are going to prove
that it cannot stray too far from being one. To this end, we need the
following simple

######  Lemma 2.

Let $(V,\Updelta)$ be a root system. There exists a unique (up to reordering)
orthogonal decomposition $V=\bigoplus_{i=1}^{k}V_{i}$ such that
$\Updelta=\bigsqcup_{i=1}^{k}\Updelta_{i}$, where $\Updelta_{i}=\Updelta\cap
V_{i}$, and $(V_{i},\Updelta_{i})$ is an irreducible root system. Two roots
$\upalpha,\upbeta\in\Updelta$ lie in the same component $\Updelta_{i}$ if and
only if there exists a chain of roots
$\uplambda_{0},\uplambda_{1},\ldots,\uplambda_{s}\in\Updelta$ with
$\uplambda_{0}=\upalpha,\uplambda_{s}=\upbeta$, such that $\langle\hskip
1.00006pt{\uplambda_{i-1}}\hskip 1.00006pt|\hskip
1.00006pt{\uplambda_{i}}\hskip 1.00006pt\rangle\hskip 0.20004pt\neq 0$ for
$1\leqslant i\leqslant s$.

Unsurprisingly, we call each $(V_{i},\Updelta_{i})$ an irreducible component
of $\boldsymbol{(V,\Updelta)}$ and the decomposition
$V=\bigoplus_{i=1}^{k}V_{i},\Updelta=\bigsqcup_{i=1}^{k}\Updelta_{i}$ the
decomposition of $\boldsymbol{(V,\Updelta)}$ into its irreducible components.

###### Proof of the lemma.

Introduce an equivalence relation on $\Updelta$: two roots
$\upalpha,\upbeta\in\Updelta$ are equivalent if and only if they can be
connected by a chain of roots
$\uplambda_{0},\uplambda_{1},\ldots,\uplambda_{s}\in\Updelta$ as above. This
is clearly an equivalence relation, so we can write
$\Updelta=\bigsqcup_{i=1}^{k}\Updelta_{i}$ for the decomposition of $\Updelta$
into the equivalence classes. Define $V_{i}$ to be the linear span of
$\Updelta_{i}$. Since $\Updelta$ spans $V$, we have $V=\sum_{i=1}^{k}V_{i}$.
By construction, given $i,j\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\},i\neq j$, every root $\upalpha\in\Updelta_{i}$ is
orthogonal to every root $\upbeta\in\Updelta_{j}$, so $V_{i}\perp V_{j}$.
Therefore, we have an orthogonal decomposition $V=\bigoplus_{i=1}^{k}V_{i}$.
In particular, this implies that $\Updelta_{i}=\Updelta\cap V_{i}$ for each
$i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$.
Trivially, for every subspace $W\subseteq V$, $(W,\Updelta\cap W)$ is a root
system, hence so is each $(V_{i},\Updelta_{i})$. Note that each $\Updelta_{i}$
is irreducible by design. Let $V=\bigoplus_{i=1}^{k^{\prime}}V^{\prime}_{i}$
be another decomposition of $V$ as in the lemma. It follows from what we have
already proven that all roots in $\Updelta\cap V^{\prime}_{i}$ are equivalent
to each other for each $i\in\left\\{\hskip
0.29999pt{1,\ldots,k^{\prime}}\hskip 0.29999pt\right\\}$. On the other hand,
if $i,j\in\left\\{\hskip 0.29999pt{1,\ldots,k^{\prime}}\hskip
0.29999pt\right\\},i\neq j$, no root in $\Updelta\cap V^{\prime}_{i}$ can be
equivalent to any root in $\Updelta\cap V^{\prime}_{j}$. Consequently, the
decomposition $V=\bigoplus_{i=1}^{k^{\prime}}V^{\prime}_{i}$ coincides with
our constructed decomposition up to reordering of the factors, which completes
the proof. ∎

Now we can prove the following result, which asserts that root system
isomorphisms are ”almost” conformal maps.

######  Proposition 3.

Let $(V,\Updelta)$ and $(V^{\prime},\Updelta^{\prime})$ be root systems and
$f\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
an isomorphism between them. Write
$V=\bigoplus_{i=1}^{k}V_{i},\Updelta=\bigsqcup_{i=1}^{k}\Updelta_{i}$ and
$V^{\prime}=\bigoplus_{i=1}^{k^{\prime}}V^{\prime}_{i},\Updelta^{\prime}=\bigsqcup_{i=1}^{k^{\prime}}\Updelta^{\prime}_{i}$
for the decompositions of $(V,\Updelta)$ and $(V^{\prime},\Updelta^{\prime})$
into their irreducible components. Then $k=k^{\prime}$ and, after reordering
$V_{i}$’s if needed, $f(V_{i})=V^{\prime}_{i}$ and
$f(\Updelta_{i})=\Updelta^{\prime}_{i}$ for each $i\in\left\\{\hskip
0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$. Moreover, for each $i$,
${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{V_{i}}}\colon
V_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}_{i}$
is a conformal map, i.e. there exists $a_{i}>0$ such that
$a_{i}{\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{V_{i}}}\colon
V_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}_{i}$
is an isometry.

###### Proof.

To begin with, observe that $\upalpha\perp\upbeta\Leftrightarrow
n_{\upalpha\upbeta}=0$, so $f$ must preserve root orthogonality. From this it
easily follows that $f$ preserves the equivalence relation on roots described
in the proof of Lemma 2, which, in turn, implies the first assertion. For the
remainder of the proof, we may assume that both $(V,\Updelta)$ and
$(V^{\prime},\Updelta^{\prime})$ are irreducible, and we need to proof that
$f$ is conformal. Pick any $\upalpha_{0}\in\Updelta$ and define
$a=\frac{||f(\upalpha_{0})||}{||\upalpha_{0}||}>0$. We will prove that
$a^{-1}f$ is an isometry. Since $a^{-1}f$ already preserves the root integers,
it suffices to show that it preserves the length of each root. Note that
$\frac{n_{\upalpha\upbeta}}{n_{\upbeta\upalpha}}=\frac{||\upalpha||^{2}}{||\upbeta||^{2}}$
whenever $\langle\hskip 1.00006pt{\upalpha}\hskip 1.00006pt|\hskip
1.00006pt{\upbeta}\hskip 1.00006pt\rangle\hskip 0.20004pt\neq 0$. Hence, $f$
preserves the length-ratio of any pair of non-orthogonal roots. Pick any
$\upbeta\in\Updelta$. According to Lemma 2, there exists a chain of roots
$\uplambda_{0},\uplambda_{1},\ldots,\uplambda_{s}$ with
$\uplambda_{0}=\upalpha_{0},\uplambda_{s}=\upbeta$, such that $\langle\hskip
1.00006pt{\uplambda_{i-1}}\hskip 1.00006pt|\hskip
1.00006pt{\uplambda_{i}}\hskip 1.00006pt\rangle\hskip 0.20004pt\neq 0$ for
$1\leqslant i\leqslant s$. We can compute:

$\frac{||f(\upalpha_{0})||}{||f(\upbeta)||}=\frac{||f(\uplambda_{0})||}{||f(\uplambda_{1})||}\frac{||f(\uplambda_{1})||}{||f(\uplambda_{2})||}\cdots\frac{||f(\uplambda_{s-1})||}{||f(\uplambda_{s})||}=\frac{||\uplambda_{0}||}{||\uplambda_{1}||}\frac{||\uplambda_{1}||}{||\uplambda_{2}||}\cdots\frac{||\uplambda_{s-1}||}{||\uplambda_{s}||}=\frac{||\upalpha_{0}||}{||\upbeta||},$

i.e
$\frac{||f(\upbeta)||}{||\upbeta||}=\frac{||f(\upalpha_{0})||}{||\upalpha_{0}||}=a$
or, in other words, $||a^{-1}f(\upbeta)||=||\upbeta||$. Consequently,
$a^{-1}f$ preserves the lengths of all the roots and is an isometry, which
means that $f$ is conformal. ∎

######  Corollary 3.1.

If $(V,\Updelta)$ is an irreducible root system, then $\mathrm{Aut}\hskip
0.59998pt(\Updelta)\subseteq\mathrm{O}(V)$.

###### Proof.

Take any automorphism $f\in\mathrm{Aut}\hskip 0.59998pt(\Updelta)$. By
Proposition 3, there exists $a>0$ such that $af$ is an isometry. Assume that
$f$ is not orthogonal, i.e. $a\neq 1$. By replacing $f$ with $f^{-1}$ if
needed, we may assume $a>1$, i.e. $f$ increases the length of any nonzero
vector. But $\Updelta$ is finite, hence so is the set of lengths of all roots
in $\Updelta$. Since $f(\Updelta)=\Updelta$, we arrive at a contradiction. ∎

Let $(V,\Updelta)$ be any root system. Let us look more closely at the
automorphism group $\mathrm{Aut}\hskip 0.59998pt(\Updelta)$. Recall that we
have the Weyl group $\mathrm{W}(\Updelta)$ generated by the reflections
$s_{\upalpha}$ in the root hyperplanes $\upalpha^{\perp}$,
$\upalpha\in\Updelta$. Each $s_{\upalpha}$ is orthogonal and preserves
$\Updelta$, hence it is an automorphism of $\Updelta$. We deduce that
$\mathrm{W}(\Updelta)\subseteq\mathrm{Aut}\hskip 0.59998pt(\Updelta)$. In
fact, it is a normal subgroup, which can be easily checked on its generators:
$f\in\mathrm{Aut}\hskip 0.59998pt(\Updelta),\upalpha\in\Updelta\Rightarrow
fs_{\upalpha}f^{-1}=s_{f(\upalpha)}$. The short exact sequence of groups
$\mathrm{W}(\Updelta)\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\Updelta)\twoheadrightarrow\mathrm{Aut}\hskip
0.59998pt(\Updelta)/\mathrm{W}(\Updelta)$ splits, albeit not canonically. In
order to split it, one first has to make a choice of positive roots. Pick a
Weyl chamber $D\subseteq V$, let $\Updelta^{+}\subseteq\Updelta$ be the
corresponding subset of positive roots and $\Uplambda=\left\\{\hskip
0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip
0.29999pt\right\\}\subseteq\Updelta^{+}$ the subset of simple roots. We denote
the corresponding Dynkin diagram by $\boldsymbol{\mathrm{DD}}$. If a simple
root $\upalpha_{i}$ has the property that $2\upalpha_{i}$ is also a root, the
corresponding vertex of the Dynkin diagram is represented by two concentrated
circles.

######  Definition 4.

Let $(V^{\prime},\Updelta^{\prime})$ be another root system with a fixed
choice of simple roots
$\Uplambda^{\prime}\subseteq\Updelta^{\prime+}\subseteq\Updelta^{\prime}$ and
the corresponding Dynkin diagram $\mathrm{DD}^{\prime}$. A bijection
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
is called a (diagram) isomorphism between $\boldsymbol{\mathrm{DD}}$ and
$\boldsymbol{\mathrm{DD}^{\prime}}$ if it a graph isomorphism that preserves
edge directions, the number of lines an edge consists of, and the number of
circles a vertex consists of. If $V^{\prime}=V,\Updelta^{\prime}=\Updelta,$
and $\Uplambda^{\prime}=\Uplambda$, we call $s$ an automorphism of
$\boldsymbol{\mathrm{DD}}$. The group of all automorphisms of $\mathrm{DD}$ is
denoted by $\boldsymbol{\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})}$.

The chief example of diagram isomorphisms comes from root system isomorphisms.
Suppose that $f\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
is an isomorphism between $\Updelta$ and $\Updelta^{\prime}$ such that
$f(\Uplambda)=\Uplambda^{\prime}$. Then
$s={\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{\Uplambda}}\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
is clearly an isomorphism between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$.
This also explains why the Dynkin diagram of a root system is well-defined in
the first place and does not depend on the choice of a Weyl chamber: if
$\Uplambda_{1}\subseteq\Updelta$ is another set of simple roots, then there
exists $w\in\mathrm{W}(\Updelta)\subseteq\mathrm{Aut}\hskip
0.59998pt(\Updelta)$ mapping $\Uplambda$ onto $\Uplambda_{1}$, so the
corresponding Dynkin diagrams $\mathrm{DD}$ and $\mathrm{DD}_{1}$ are
isomorphic. This construction
($f\mapsto{\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{\Uplambda}}$) actually
exhausts111This implies that a root system is fully determined by its Dynkin
diagram up to isomorphism. all Dynkin diagram isomorphisms between
$\mathrm{DD}$ and $\mathrm{DD}^{\prime}$. Although this is a standard fact in
the theory of root systems (see, for example, [7, Prop. 2.66]), we will
reprove it for our own purposes in Proposition 6 below.

Recall that for each $r\geqslant 1$ there exists only one irreducible
nonreduced root system of rank $r$ up to isomorphism (see [7, Prop. 2.92]). It
is denoted by $(BC)_{r}$ and its Dynkin diagram looks like this:

[arrow style=length=5pt, scale=3]B O5

######  Remark 5.

Some authors who work only with reduced root systems ask
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
in Definition 4 to preserve the Cartan matrix instead. This is equivalent to
our definition for reduced root systems, as the Cartan matrix and the Dynkin
diagram encode the same amount of data for such systems. However, for
nonreduced root systems, our definition is stronger because the Dynkin diagram
carries more (in fact, all) information about the root system in this case.
For instance, the Cartan matrices of $B_{r}$ and $(BC)_{r}$ are the same,
whereas their Dynkin diagrams are not – the difference is precisely the vertex
represented by two concentrated circles.

It is very straightforward to compute the group $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ for all irreducible root systems by looking at their
classification:

$\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})\simeq\begin{cases}S_{3},&\text{if}\hskip
3.99994pt\Updelta\simeq D_{4};\\\ \mathbb{Z}/2\mathbb{Z},&\text{if}\hskip
3.99994pt\Updelta\simeq A_{n}\,(n\geqslant 2),D_{n}\,(n\geqslant 5),\hskip
1.00006pt\text{or}\hskip 1.99997ptE_{6};\\\ \left\\{\hskip 0.29999pt{e}\hskip
0.29999pt\right\\},&\text{otherwise}.\end{cases}$

Since the set of simple roots forms a basis for the underlying space of a root
system, every diagram isomorphism
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$ extends uniquely to a linear
isomorphism
$V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$,
which we denote by the same letter. In particular, we have a natural group
embedding $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})\subseteq\mathrm{GL}(V)$
(once again, this embedding only makes sense after we fix the set of simple
roots). Before we relate diagram isomorphisms to root system isomorphisms, we
make a few observations. First off, note that $\mathrm{Aut}\hskip
0.59998pt(\Updelta)$ acts naturally on the set of Weyl chambers of
$(V,\Updelta)$. Second, let
$V=\bigoplus_{i=1}^{k}V_{i},\Updelta=\bigsqcup_{i=1}^{k}\Updelta_{i}$ be the
decomposition of $(V,\Updelta)$ into its irreducible components. It is easy to
see that for each $i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\}$, $\Updelta^{+}_{i}=\Updelta_{i}\cap\Updelta^{+}$ is a set
of positive roots for $\Updelta_{i}$. Consequently, we have
$\Uplambda=\bigsqcup_{i=1}^{k}\Uplambda_{i}$ and $D=\prod_{i=1}^{k}D_{i}$,
where $\Uplambda_{i}=\Uplambda\cap\Updelta_{i}^{+}$ is a set of simple roots
for $\Updelta_{i}$ and $D_{i}=D\cap V_{i}$ is the corresponding Weyl chamber.
This implies that for each $i$, the Dynkin diagram $\mathrm{DD}_{i}$ of
$\Updelta_{i}$ is a connected component of $\mathrm{DD}$, and we have
$\mathrm{DD}=\bigsqcup_{i=1}^{k}\mathrm{DD}_{i}$. Finally, note that
$\mathrm{W}(\Updelta)=\prod_{i=1}^{k}\mathrm{W}(\Updelta_{i})$.

######  Proposition 6.

Let $(V,\Updelta)$ and $(V^{\prime},\Updelta^{\prime})$ be root systems with
fixed choices of simple roots $\Uplambda\subseteq\Updelta$ and
$\Uplambda^{\prime}\subseteq\Updelta^{\prime}$.

  1. (1)

Given any diagram isomorphism
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$, its linear extension
$s\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
is an isomorphism between $\Updelta$ and $\Updelta^{\prime}$. An isomorphism
$V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
between $\Updelta$ and $\Updelta^{\prime}$ comes from a diagram isomorphism
$\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
precisely when it maps $\Uplambda$ onto $\Uplambda^{\prime}$.

  2. (2)

$\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})\subseteq\mathrm{Aut}\hskip
0.59998pt(\Updelta)$. In terms of the action of $\mathrm{Aut}\hskip
0.59998pt(\Updelta)$ on the set of Weyl chambers, $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ is the stabilizer of $D$.

  3. (3)

$\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})=\mathrm{W}(\Updelta)\rtimes\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$.

###### Proof.

Let
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
be a diagram isomorphism between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$.
Recall that the Weyl group of a root system is generated by the simple
reflections with respect to any choice of simple roots: $\mathrm{W}(\Updelta)$
is generated by $\left\\{\hskip
0.29999pt{s_{\upalpha}\mid\upalpha\in\Uplambda}\hskip 0.29999pt\right\\}$ and
the same is true for $\mathrm{W}(\Updelta^{\prime})$. Since
$s(\Uplambda)=\Uplambda^{\prime}$, we deduce that
$s\mathrm{W}(\Updelta)s^{-1}=\mathrm{W}(\Updelta^{\prime})$. On the other
hand, it is well known that every root in a root system is simple (or double
of a simple one) for a suitable choice of a Weyl chamber ([7, Prop. 2.62]).
Since the Weyl group acts transitively on the set of Weyl chambers, we deduce
that
$\Updelta=\mathrm{W}(\Updelta)\cdot(\Uplambda\cup(2\Uplambda\cap\Updelta))$
(the same is true for $\Updelta^{\prime})$. We know that for any
$\upalpha\in\Uplambda$, $2\upalpha$ is a root if and only if $2s(\upalpha)$ is
one. Altogether, we have:

$s(\Updelta)=s(\mathrm{W}(\Updelta)\cdot(\Uplambda\cup(2\Uplambda\cap\Updelta)))=s\mathrm{W}(\Updelta)s^{-1}\cdot
s(\Uplambda\cup(2\Uplambda\cap\Updelta))=\mathrm{W}(\Updelta^{\prime})\cdot(\Uplambda^{\prime}\cup(2\Uplambda^{\prime}\cap\Updelta^{\prime}))=\Updelta^{\prime},$

so $s$ satisfies condition (i) of Definition 1. What for condition (ii),
observe that $s$ provides a bijection between the connected components of
$\mathrm{DD}$ and those of $\mathrm{DD}^{\prime}$. Thus, for each
$i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$, there
exists $j\in\left\\{\hskip 0.29999pt{1,\ldots,k^{\prime}}\hskip
0.29999pt\right\\}$ (clearly, $k^{\prime}=k$) such that
$s(\mathrm{DD}_{i})=\mathrm{DD}^{\prime}_{j}$, which means that
$s(\Uplambda_{i})=\Uplambda^{\prime}_{j}$ and thus $s(V_{i})=V^{\prime}_{j}$
and $s(\Updelta_{i})=\Updelta^{\prime}_{j}$. Take any
$\upalpha\in\Uplambda_{i}$ and let $a=\frac{||s(\upalpha)||}{||\upalpha||}$.
We want to show that for every other $\upbeta\in\Uplambda_{i}$,
$\frac{||s(\upbeta)||}{||\upbeta||}=a$. Assume that $\upbeta$ is connected to
$\upalpha$ by an edge. Consider the root systems
$\Updelta\cap\mathrm{span}_{\mathbb{R}}\left\\{\hskip
0.29999pt{\upalpha,\upbeta}\hskip 0.29999pt\right\\}$ and
$\Updelta^{\prime}\cap\mathrm{span}_{\mathbb{R}}\left\\{\hskip
0.29999pt{s(\upalpha),s(\upbeta)}\hskip 0.29999pt\right\\}$. They are both of
rank 2 and we have an isomorphism between their Dynkin diagrams provided by
$s$. Since there are just five root systems of rank 2 up to isomorphism, it is
straightforward to see that two such root systems with isomorphic Dynkin
diagrams are isomorphic. What it means for us is that
$\frac{||\upbeta||}{||\upalpha||}=\frac{||s(\upbeta)||}{||s(\upalpha)||}$,
hence
$\frac{||s(\upbeta)||}{||\upbeta||}=\frac{||s(\upalpha)||}{||\upalpha||}=a$.
Since $\mathrm{DD}_{i}$ is connected, it follows by induction that $s$
increases the lengths of all simple roots in $\Uplambda_{i}$ by the same
factor of $a$. As we already know that it preserves the Cartan integers, we
deduce that it is conformal on $V_{i}$ ($a^{-1}s\colon
V_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}_{j}$
is an isometry). But this, together with condition (i) of Definition 1,
implies that it preserves the root integers between all the roots in
$\Updelta_{i}$ (and not only between the simple ones). Since the root integers
between roots lying in different components of $\Updelta$ are all zero, we see
that $s$ is a root system isomorphism, which was to be proven. The second
assertion in part (1) of the proposition is trivial.

Part (2) follows from part (1), as $s\in\mathrm{Aut}\hskip
0.59998pt(\Updelta)$ preserves $D$ if and only if it preserves $\Uplambda$.

Part (3) hinges on the fact that $\mathrm{W}(\Updelta)$ acts simply
transitively on the set of Weyl chambers. It is clear from (2) that
$\mathrm{W}(\Updelta)$ and $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})$ do not
intersect. On the other hand, let $f\in\mathrm{Aut}\hskip
0.59998pt(\Updelta)$. There exists $w\in\mathrm{W}(\Updelta)$ such that
$w(f(D))=D$. But then $s=wf$ fixes $D$ and thus lies in $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$. Therefore, we have a decomposition
$f=w^{-1}s,w^{-1}\in\mathrm{W}(\Upsigma),s\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$. This completes the proof of part (3). ∎

Finally, we discuss some aspects of the correspondence between reduced root
systems and complex semisimple Lie algebras. Let $\mathfrak{{g}}$ be a
(finite-dimensional) complex semisimple Lie algebra. Pick a Cartan subalgebra
$\mathfrak{{h}}\subset\mathfrak{{g}}$. We have the corresponding set of roots
$\Updelta\subset\mathfrak{{h}}^{*}$ and the root space decomposition
$\mathfrak{{g}}=\mathfrak{{h}}\oplus\bigoplus_{\upalpha\in\Updelta}\mathfrak{{g}}_{\upalpha}$.
The restriction of the Killing form $B$ of $\mathfrak{{g}}$ to
$\mathfrak{{h}}$ is nondegenerate, so it induces a $\mathbb{C}$-linear
isomorphism
$\mathfrak{{h}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{*}$.
Write $\mathfrak{{h}}^{*}(\mathbb{R})\subset\mathfrak{{h}}^{*}$ for the real
span of $\Updelta$ and $\mathfrak{{h}}(\mathbb{R})\subset\mathfrak{{h}}$ for
its preimage under
$\mathfrak{{h}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{*}$.
It is a standard fact that $\mathfrak{{h}}(\mathbb{R})=\left\\{\hskip
0.29999pt{h\in\mathfrak{{h}}\mid f(h)\in\mathbb{R}\hskip 5.0pt\forall\hskip
1.0ptf\in\mathfrak{{h}}^{*}(\mathbb{R})}\hskip 0.29999pt\right\\}$ (hence
$\mathfrak{{h}}^{*}(\mathbb{R})$ is the real dual of
$\mathfrak{{h}}(\mathbb{R})$), and we have
$\mathfrak{{h}}=\mathfrak{{h}}(\mathbb{R})\oplus_{\mathbb{R}}i\mathfrak{{h}}(\mathbb{R})$
and
$\mathfrak{{h}}^{*}=\mathfrak{{h}}^{*}(\mathbb{R})\oplus_{\mathbb{R}}i\mathfrak{{h}}^{*}(\mathbb{R})$.
The restriction of $B$ to $\mathfrak{{h}}(\mathbb{R})$ is positive definite
and we can carry it along the isomorphism
$\mathfrak{{h}}(\mathbb{R})\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{*}(\mathbb{R})$
to an inner product on $\mathfrak{{h}}^{*}(\mathbb{R})$. This makes
$(\mathfrak{{h}}^{*}(\mathbb{R}),\Updelta)$ into a reduced root system. Note
that this inner product on $\mathfrak{{h}}^{*}(\mathbb{R})$ is natural and
does not require any additional choices, for it comes from the Killing form,
which is fully determined by Lie algebra structure of $\mathfrak{{g}}$.

Now we make a choice of positive roots $\Updelta^{+}\subset\Updelta$ and let
$\Uplambda=\left\\{\hskip 0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip
0.29999pt\right\\}$ be the corresponding set of simple roots. Write
$H_{i}\in\mathfrak{{h}}(\mathbb{R})$ for the preimage of $\upalpha_{i}$ under
the isomorphism
$\mathfrak{{h}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{*}$
and let $h_{i}=\frac{2}{||\upalpha_{i}||^{2}}H_{i}$. Finally, make a choice of
canonical generators
$e_{i}\in\mathfrak{{g}}_{\upalpha_{i}},f_{i}\in\mathfrak{{g}}_{-\upalpha_{i}}$.
It follows from the definition of $h_{i}$’s that

$[h_{i},e_{j}]=n_{\upalpha_{j}\upalpha_{i}}e_{j},\quad[h_{i},f_{j}]=-n_{\upalpha_{j}\upalpha_{i}}f_{j}.$

The Isomorphism Theorem asserts that if $\mathfrak{{g}}^{\prime}$ is another
complex semisimple Lie algebra with a fixed choice of
$\mathfrak{{h}}^{\prime},\,\Uplambda^{\prime}=\left\\{\hskip
0.29999pt{\upalpha^{\prime}_{1},\ldots,\upalpha^{\prime}_{r}}\hskip
0.29999pt\right\\}\subset\Updelta^{\prime},$ and
$e^{\prime}_{i}\in\mathfrak{{g}}_{\upalpha^{\prime}_{i}},\,f^{\prime}_{i}\in\mathfrak{{g}}_{-\upalpha^{\prime}_{i}},\,1\leqslant
i\leqslant r$, such that the Cartan matrices
$A=(n_{\upalpha_{i}\upalpha_{j}})_{i,j=1}^{r}$ and
$A^{\prime}=(n_{\upalpha^{\prime}_{i}\upalpha^{\prime}_{j}})_{i,j=1}^{r}$
coincide, then there exists a unique Lie algebra isomorphism
$\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
sending $h_{i}$ to $h^{\prime}_{i}$, $e_{i}$ to $e^{\prime}_{i}$, and $f_{i}$
to $f^{\prime}_{i}$ for $1\leqslant i\leqslant r$.

Let
$F\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
be a Lie algebra isomorphism mapping $\mathfrak{{h}}$ onto
$\mathfrak{{h}}^{\prime}$ and write
$f=({\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{\mathfrak{{h}}}}^{*})^{-1}\colon\mathfrak{{h}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}$.
It is a matter of simple computation that $f(\Updelta)=\Updelta^{\prime}$ and
for any $\upalpha\in\Updelta$,
$F(\mathfrak{{g}}_{\upalpha})=\mathfrak{{g}}^{\prime}_{f(\upalpha)}$. In
particular, we have
$f(\mathfrak{{h}}^{*}(\mathbb{R}))=\mathfrak{{h}}^{\prime*}(\mathbb{R})$. We
will slightly abuse the notation and use the same letter $f$ for the
restriction
${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{\mathfrak{{h}}^{*}(\mathbb{R})}}\colon\mathfrak{{h}}^{*}(\mathbb{R})\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}(\mathbb{R})$.
As $F$ respects the Killing forms of $\mathfrak{{g}}$ and
$\mathfrak{{g}}^{\prime}$, it follows that $f$ is an isometry and thus a root
system isomorphism222This construction also shows that different choices of a
Cartan subalgebra of $\mathfrak{{g}}$ lead to isomorphic root systems, as any
two Cartan subalgebras differ by an inner automorphism of $\mathfrak{{g}}$.
The map sending $\mathfrak{{g}}$ to
$(\mathfrak{{h}}^{*}(\mathbb{R}),\Updelta)$ is a 1-to-1 correspondence between
the isomorphism classes of complex semisimple Lie algebras on the one hand and
of reduced root systems on the other. between $\Updelta$ and
$\Updelta^{\prime}$. The Isomorphism Theorem ensures that every isomorphism
$\mathfrak{{h}}^{*}(\mathbb{R})\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}(\mathbb{R})$
between $\Updelta$ and $\Updelta^{\prime}$ arises in this way (this fact is
essentially equivalent to the Isomorphism Theorem and is proven directly in
[7, Th. 2.108]). As a consequence, every such isomorphism is an isometry.

If we let $\mathfrak{{g}}^{\prime}=\mathfrak{{g}}$ and
$\mathfrak{{h}}^{\prime}=\mathfrak{{h}}$, we get a surjective Lie group
homomorphism $\boldsymbol{\Uppsi}\colon N_{\mathrm{Aut}\hskip
0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})\twoheadrightarrow\mathrm{Aut}\hskip
0.59998pt(\Updelta),\,F\mapsto f$, where $N_{\mathrm{Aut}\hskip
0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ is the normalizer of
$\mathfrak{{h}}$ in $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$.

For each $\upalpha\in\Updelta$, an automorphism $\upeta\in
N_{\mathrm{Aut}\hskip 0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ such that
$\Uppsi(\upeta)=s_{\upalpha}$ can be constructed explicitly (here
$s_{\upalpha}$ is the reflection of $\mathfrak{{h}}^{*}(\mathbb{R})$ in the
hyperplane $\upalpha^{\perp}$). Namely, if
$e_{\upalpha}\in\mathfrak{{g}}_{\upalpha}$ and
$f_{\upalpha}\in\mathfrak{{g}}_{-\upalpha}$ are such that
$[e_{\upalpha},f_{\upalpha}]=h_{\upalpha}$ ($\Leftrightarrow
B(e_{\upalpha},f_{\upalpha})=\frac{2}{||\upalpha||^{2}}$), then
$\upeta=\exp(\mathrm{ad}\hskip
0.59998pt\frac{\uppi}{2}(e_{\upalpha}-f_{\upalpha}))\in N_{\mathrm{Aut}\hskip
0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ and $\Uppsi(\upeta)=s_{\upalpha}$
(see [4, p. 210]). Observe also that the Isomorphism Theorem allows to
construct a section $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})\hookrightarrow
N_{\mathrm{Aut}\hskip 0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ of $\Uppsi$
over $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})$: take $s\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ and send it to $\hat{s}\in N_{\mathrm{Aut}\hskip
0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ given by
$\hat{s}(h_{i})=h_{s(i)},\hat{s}(e_{i})=e_{s(i)},\hat{s}(f_{i})=f_{s(i)}$
(here we think of $s$ as a permutation of $\left\\{\hskip
0.29999pt{1,\ldots,r}\hskip 0.29999pt\right\\}\simeq\left\\{\hskip
0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip 0.29999pt\right\\}$).

Eventually, we make the following observation. Recall that a root system
isomorphism is not in general an isometry (or even a conformal map). However,
as we have seen, if $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ are complex
semisimple Lie algebras with Cartan subalgebras $\mathfrak{{h}}$ and
$\mathfrak{{h}}^{\prime}$, respectively, then every root system isomorphism
$\mathfrak{{h}}^{*}(\mathbb{R})\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}(\mathbb{R})$
between $\Updelta$ and $\Updelta^{\prime}$ is an isometry. The reason for this
is that the inner products on $\mathfrak{{h}}^{*}(\mathbb{R})$ and
$\mathfrak{{h}}^{\prime*}(\mathbb{R})$ are ”nice” – they come from the Killing
forms of $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$. In fact, we can make
the following definition. Let $(V,\Updelta)$ be any reduced root system. Take
a complex semisimple Lie algebra $\mathfrak{{g}}$ and a Cartan subalgebra
$\mathfrak{{h}}\subset\mathfrak{{g}}$ such that the root system
$(\mathfrak{{h}}^{*}(\mathbb{R}),\Updelta_{\mathfrak{{g}}})$ is isomorphic to
$(V,\Updelta)$. Pick any isomorphism $\upvarphi\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{*}(\mathbb{R})$
between $\Updelta$ and $\Updelta_{\mathfrak{{g}}}$ and carry the inner product
from $\mathfrak{{h}}^{*}(\mathbb{R})$ to $V$ along $\upvarphi$ (as we know
from Proposition 3, it simply amounts to renormalizing the existing inner
product on $V$ by some conformal factors on the irreducible components of
$(V,\Updelta)$). Suppose we have another pair
$(\mathfrak{{g}}^{\prime},\mathfrak{{h}}^{\prime})$ and an isomorphism
$\upvarphi^{\prime}\colon
V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}(\mathbb{R})$
between $\Updelta$ and $\Updelta_{\mathfrak{{g}}^{\prime}}$. We claim that the
inner product on $V$ pulled back from $\mathfrak{{h}}^{\prime*}(\mathbb{R})$
along $\upvarphi^{\prime}$ is the same. Indeed, if we write
$f=\upvarphi^{\prime}\circ\upvarphi^{-1}\colon\mathfrak{{h}}^{*}(\mathbb{R})\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{h}}^{\prime*}(\mathbb{R})$,
then $f$ is an isomorphism between $\Updelta_{\mathfrak{{g}}}$ and
$\Updelta_{\mathfrak{{g}}^{\prime}}$. As we discussed above, each such
isomorphism is an isometry, which proves the claim. We call the inner product
on $V$ constructed above Killing.

######  Corollary 7.

Let $(V,\Updelta)$ and $(V^{\prime},\Updelta^{\prime})$ be reduced root
systems with Killing inner products. Then every isomorphism
$V\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}V^{\prime}$
between $\Updelta$ and $\Updelta^{\prime}$ is an isometry. In particular,
$\mathrm{Aut}\hskip 0.59998pt(\Updelta)\subseteq\mathrm{O}(V)$.

##  3\. Real semisimple Lie algebras and their restricted root systems

In this section we discuss restricted root systems of real semisimple Lie
algebras and prove the main results. See [7, Ch. VI] and [10, §2,3] for a
detailed exposition of the theory of real semisimple Lie algebras.

Let $\mathfrak{{g}}$ be a (finite-dimensional) real semisimple Lie algebra.
Let us fix a Cartan involution $\uptheta$ of $\mathfrak{{g}}$ and write
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$ for the corresponding
Cartan decomposition. We also pick a maximal abelian subspace $\mathfrak{{a}}$
of $\mathfrak{{p}}$. Having this fixed, we have the set of restricted roots
$\Upsigma\subseteq\mathfrak{{a}}^{*}$ and the restricted root space
decomposition
$\mathfrak{{g}}=\mathfrak{{g}}_{0}\oplus\bigoplus_{\upalpha\in\Upsigma}\mathfrak{{g}}_{\upalpha}$,
where
$\mathfrak{{g}}_{0}=Z_{\mathfrak{{g}}}(\mathfrak{{a}})=\mathfrak{{k}}_{0}\oplus\mathfrak{{a}}$
and
$\mathfrak{{k}}_{0}=Z_{\mathfrak{{k}}}(\mathfrak{{a}})=N_{\mathfrak{{k}}}(\mathfrak{{a}})$.
Note also that
$\uptheta(\mathfrak{{g}}_{\upalpha})=\mathfrak{{g}}_{-\upalpha}$ for any
$\upalpha\in\Upsigma$. If we write $B$ for the Killing form of
$\mathfrak{{g}}$, then $B_{\uptheta}(X,Y)=-B(X,\uptheta Y)$ is an inner
product on $\mathfrak{{g}}$ (which we choose as out default inner product on
$\mathfrak{{g}}$ unless otherwise specified). The summands of the restricted
root space decomposition are all pairwise orthogonal with respect to
$B_{\uptheta}$, and $\mathfrak{{k}}$ and $\mathfrak{{p}}$ are mutually
orthogonal with respect to both $B$ and $B_{\uptheta}$. Note that
$B_{\uptheta}$ coincides with $B$ on $\mathfrak{{p}}$ and equals $-B$ on
$\mathfrak{{k}}$. In a standard fashion,
${\left.\kern-1.2ptB_{\uptheta}\vphantom{\big{|}}\right|_{\mathfrak{{a}}\times\mathfrak{{a}}}}={\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{a}}\times\mathfrak{{a}}}}$
induces an isomorphism
$\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$,
and we carry it along this isomorphism to an inner product on
$\mathfrak{{a}}^{*}$. This way, $(\mathfrak{{a}}^{*},\Upsigma)$ becomes a root
system, called the restricted root system of $\mathfrak{{g}}$. There are three
main differences with the complex semisimple case (apart from the obvious
difference in how the root system is constructed):

  1. (1)

$\Upsigma$ does not have to be reduced.

  2. (2)

$\Upsigma$ loses all information about the compact ideals of $\mathfrak{{g}}$.

  3. (3)

The dimensions of the restricted root subspaces $\mathfrak{{g}}_{\upalpha}$ do
not have to be equal to 1.

Let us address these points individually. To begin with, (1) is not really an
issue, since we know the classification of all – not necessarily reduced –
root systems up to isomorphism. As we mentioned in Section 2, the root systems
$(BC)_{r},\,r\geqslant 1$, exhaust the list of all irreducible nonreduced root
systems up to isomorphism.

Regarding (2), we make the following observation. First of all, it is a
standard fact about Cartan involutions that $\mathfrak{{k}}$ is a maximal
compact subalgebra of $\mathfrak{{g}}$. This immediately implies that

$\mathfrak{{g}}\hskip 4.0pt\text{is
compact}\,\Leftrightarrow\;\mathfrak{{p}}=\left\\{\hskip 0.29999pt{0}\hskip
0.29999pt\right\\}\;\Leftrightarrow\;\mathfrak{{a}}=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}\;\Leftrightarrow\;\Upsigma=\varnothing.$

Now, let $\mathfrak{{g}}=\bigoplus_{i=1}^{k}\mathfrak{{g}}_{i}$ be the
decomposition of $\mathfrak{{g}}$ into the sum of its simple ideals.
Necessarily, $\uptheta=\uptheta_{1}+\cdots+\uptheta_{k}$, where each
$\uptheta_{i}$ is a Cartan involution on $\mathfrak{{g}}_{i}$. We then have
$\mathfrak{{k}}=\bigoplus_{i=1}^{k}\mathfrak{{k}}_{i},\,\mathfrak{{p}}=\bigoplus_{i=1}^{k}\mathfrak{{p}}_{i}$,
where
$\mathfrak{{k}}_{i}=\mathfrak{{k}}\cap\mathfrak{{g}}_{i},\,\mathfrak{{p}}_{i}=\mathfrak{{p}}\cap\mathfrak{{g}}_{i}$.
Furthermore, we necessarily have
$\mathfrak{{a}}=\bigoplus_{i=1}^{k}\mathfrak{{a}}_{i},\,\mathfrak{{a}}_{i}=\mathfrak{{a}}\cap\mathfrak{{p}}_{i}$,
and hence
$\mathfrak{{a}}^{*}=\bigoplus_{i=1}^{k}\mathfrak{{a}}_{i}^{*},\,\Upsigma=\bigsqcup_{i=1}^{k}\Upsigma_{i}$,
where $\Upsigma_{i}=\Upsigma\cap\mathfrak{{a}}_{i}^{*}$ is the restricted root
system of $\mathfrak{{g}}_{i}$. Observe also that the decomposition
$\mathfrak{{g}}=\bigoplus_{i=1}^{k}\mathfrak{{g}}_{i}$ is orthogonal with
respect to the inner product $B_{\uptheta}$, and
$B_{\uptheta}=B_{\uptheta_{1}}^{1}+\cdots+B_{\uptheta_{k}}^{k}$, where $B^{i}$
is the Killing form of $\mathfrak{{g}}_{i}$. Consequently, the decomposition
$\mathfrak{{a}}^{*}=\bigoplus_{i=1}^{k}\mathfrak{{a}}_{i}^{*}$ is orthogonal
as well. It is fairly easy to see from the properties of the restricted root
space decomposition that a simple Lie algebra must have an irreducible
restricted root system. We deduce that
$\Upsigma=\Upsigma_{1}\sqcup\ldots\sqcup\Upsigma_{k}$ is the decomposition of
$\Upsigma$ into its irreducible components (except for the fact that some
components $\Upsigma_{i}$ might be empty – precisely when the correspondent
simple ideal $\mathfrak{{g}}_{i}$ is compact). We see that it is impossible to
recover information about the compact ideals of $\mathfrak{{g}}$ from
$\Upsigma$. We also arrive at the following

######  Corollary 8.

Let $\mathfrak{{g}}=\bigoplus_{i=1}^{k}\mathfrak{{g}}_{i}$ as above. Then
$\Upsigma$ is irreducible if and only if at most one simple ideal
$\mathfrak{{g}}_{i}$ is noncompact. In particular, if $\mathfrak{{g}}$ has no
nonzero compact ideals and $\Upsigma$ is irreducible, then $\mathfrak{{g}}$ is
simple.

Let us now address (3) – arguably, the most important difference with the
complex semisimple case. We make use of the fact that
$\dim_{\mathbb{R}}(\mathfrak{{g}}_{\upalpha})$ might not be equal to $1$ and
call it the multiplicity of the root $\upalpha$ (and denote it by
$\boldsymbol{\mathrm{mult}(\upalpha)}$). The idea is that we incorporate this
information into the root system $\Upsigma$ itself. To this end, we make the
following definition: a weighted root system is a root system in which every
root is assigned a positive integer, called the multiplicity of the root444In
this paper, we are using the letter $\Updelta$ for regular root systems and
reserve $\Upsigma$ for weighted ones.. The assignment of multiplicities to the
roots in $\Upsigma$ is far from random, but since real semisimple Lie algebras
seem to be the only context where root multiplicities arise, we do not make
any additional assumptions in the definition. Since
$\uptheta(\mathfrak{{g}}_{\upalpha})=\mathfrak{{g}}_{-\upalpha}$, we know that
$\mathrm{mult}(\upalpha)=\mathrm{mult}(-\upalpha)$ for any
$\upalpha\in\Upsigma$. Below we will see that much more is true (see Theorem
15(2) and Proposition 16(5)).

######  Definition 9.

Let $\mathfrak{{g}}^{\prime}$ be another real semisimple Lie algebra with a
Cartan involution $\uptheta^{\prime}$ and a maximal abelian subspace
$\mathfrak{{a}}^{\prime}\subseteq\mathfrak{{p}}^{\prime}$ fixed, and let
$\Upsigma^{\prime}\subseteq\mathfrak{{a}}^{\prime*}$ be the corresponding
restricted root system. We call a root system isomorphism
$f\colon\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime*}$
between $\Upsigma$ and $\Upsigma^{\prime}$ weighted if it preserves the root
multiplicities: $\mathrm{mult}(f(\upalpha))=\mathrm{mult}(\upalpha)$ for every
$\upalpha\in\Upsigma$. We say that $(\mathfrak{{a}}^{*},\Upsigma)$ and
$(\mathfrak{{a}}^{\prime*},\Upsigma^{\prime})$ are weighted-isomorphic if
there exists a weighted isomorphism between them. Finally, we call a weighted
isomorphism
$\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$
from $\Upsigma$ to itself a weighted automorphism of
$\boldsymbol{(\mathfrak{{a}}^{*},\Upsigma)}$ (or of $\boldsymbol{\Upsigma}$,
for short). The group of all weighted automorphisms of
$(\mathfrak{{a}}^{*},\Upsigma)$ will be denoted by
$\boldsymbol{\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)}\subseteq\mathrm{Aut}\hskip
0.59998pt(\Upsigma)$.

We want to relate weighted root system isomorphisms to Lie algebra
isomorphisms. Let $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ be as above
and suppose that
$F\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
is an isomorphism such that $F\circ\uptheta=\uptheta^{\prime}\circ F$ (hence
$F(\mathfrak{{k}})=\mathfrak{{k}}^{\prime},F(\mathfrak{{p}})=\mathfrak{{p}}^{\prime}$)
and $F(\mathfrak{{a}})=\mathfrak{{a}}^{\prime}$. Consider
${\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}\colon\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime}$
and define
$f=({\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}^{*})^{-1}\colon\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime*}$.
Similarly to what we did in the complex semisimple case, it is easy to check
that $f(\Upsigma)=\Upsigma^{\prime}$ and
$F(\mathfrak{{g}}_{\upalpha})=\mathfrak{{g}}^{\prime}_{f(\upalpha)}$ for every
$\upalpha\in\Upsigma$. It is also clear that $F$ is an isometry with respect
to the inner products $B_{\uptheta}$ and $B^{\prime}_{\uptheta^{\prime}}$, so
$f$ is an isometry as well. All this implies that $f$ is a weighted
isomorphism between $\Upsigma$ and $\Upsigma^{\prime}$.

We can apply this construction to the situation when
$\mathfrak{{g}}^{\prime}=\mathfrak{{g}},\uptheta^{\prime}=\uptheta,$ and
$\mathfrak{{a}}^{\prime}=\mathfrak{{a}}$. Consider the Lie group
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$. We have a distinguished
element of this group fixed, namely the Cartan involution
$\uptheta\in\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$. Consider the closed
subgroup $K=Z_{\mathrm{Aut}\hskip 0.41998pt(\mathfrak{{g}})}(\uptheta)$ of
automorphisms that commute with $\uptheta$. It can also be described as the
fixed point subgroup $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{\uptheta}$
of the involutive automorphism
$C_{\uptheta}\colon\upeta\mapsto\uptheta\upeta\uptheta^{-1}=\uptheta\upeta\uptheta$
of $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ (strictly speaking, we
should write $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{C_{\uptheta}}$,
not $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{\uptheta}$, but this is a
common abuse of notation). Yet another way to describe it is
$K=N_{\mathrm{Aut}\hskip 0.41998pt(\mathfrak{{g}})}(\mathfrak{{k}})$, the
normalizer of $\mathfrak{{k}}$ in $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$. Indeed, an automorphism $\upeta$ commutes with
$\uptheta$ if and only if it respects the Cartan decomposition
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$. Since every automorphism
of $\mathfrak{{g}}$ is orthogonal with respect to the Killing form $B$ and
$\mathfrak{{p}}$ is the orthogonal complement of $\mathfrak{{k}}$ with respect
to $B$, $\upeta(\mathfrak{{k}})=\mathfrak{{k}}$ automatically implies
$\upeta(\mathfrak{{p}})=\mathfrak{{p}}$. As we will see later, $K$ is a
maximal compact subgroup of $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$
(see Corollary 26). Since $\mathfrak{{g}}$ is semisimple, we can identify it
with the Lie algebra of $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ by
means of the adjoint representation: $\mathrm{ad}\hskip
0.59998pt\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{Der}(\mathfrak{{g}})=\mathrm{Lie}\hskip
0.59998pt(\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}}))$. Under this
identification, we have

$\displaystyle\mathrm{Lie}\hskip 0.59998pt(K)$
$\displaystyle=\mathrm{Der}(\mathfrak{{g}})^{\mathrm{Ad}\hskip
0.41998pt(\uptheta)}$ $\displaystyle\cong\left\\{\hskip
0.29999pt{X\in\mathfrak{{g}}\mid\mathrm{ad}\hskip
0.59998pt(X)\circ\uptheta=\uptheta\circ\mathrm{ad}\hskip 0.59998pt(X)}\hskip
0.29999pt\right\\}$ $\displaystyle=\left\\{\hskip
0.29999pt{X\in\mathfrak{{g}}\mid\mathrm{ad}\hskip
0.59998pt(X)=\mathrm{ad}\hskip 0.59998pt(\uptheta(X))}\hskip
0.29999pt\right\\}$ $\displaystyle=\left\\{\hskip
0.29999pt{X\in\mathfrak{{g}}\mid\uptheta(X)=X}\hskip
0.29999pt\right\\}=\mathfrak{{k}}.$

One can show that $K=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})\cap\mathrm{O}_{B_{\uptheta}}(\mathfrak{{g}})$ (see
[5, Lem. 2.2]). We are especially interested in the subgroup
$N_{K}(\mathfrak{{a}})$ of $K$. According to the previous paragraph, we can
define a map $\Upomega\colon N_{K}(\mathfrak{{a}})\to\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma),\upvarphi\mapsto({\left.\kern-1.2pt\upvarphi\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}^{*})^{-1}$.
This is easily seen to be a Lie group homomorphism.

Using some standard facts from the theory of real semisimple Lie algebras, we
can prove the following:

######  Proposition 10.

If $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ are isomorphic real
semisimple Lie algebras, then their restricted root systems are weighted-
isomorphic for any choices of $\uptheta,\uptheta^{\prime},\mathfrak{{a}},$ and
$\mathfrak{{a}}^{\prime}$. In particular, the restricted root system of
$\mathfrak{{g}}$ does not depend on the choice of $\uptheta$ and
$\mathfrak{{a}}$ (up to a weighted isomorphism).

###### Proof.

Let
$F\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
be any isomorphism. Since any two Cartan involutions on $\mathfrak{{g}}$ are
conjugate by an inner automorphism, we may assume without loss of generality
that $F\circ\uptheta=\uptheta^{\prime}\circ F$. Moreover, any two maximal
abelian subspaces of $\mathfrak{{p}}$ differ by an element of $K$ (even of
$K^{0}$; see [7, Th. 6.51]). Since, by definition, any element of $K$ commutes
with $\uptheta$, we may assume, in addition, that
$F(\mathfrak{{a}})=\mathfrak{{a}}^{\prime}$. But now the above construction
implies that $F$ induces a weighted isomorphism between $\Upsigma$ and
$\Upsigma^{\prime}$. ∎

In a similar vein to the complex semisimple case, one can show that
$\mathrm{W}(\Upsigma)\subseteq\hskip 0.80002pt\mathrm{Im}\hskip
0.39993pt(\Upomega)$. In fact, if $\upalpha\in\Upsigma$ and
$X\in\mathfrak{{g}}_{\upalpha}$ is a vector of length
$\frac{\sqrt{2}}{||\upalpha||}$, then $\exp(\mathrm{ad}\hskip
0.59998pt\frac{\uppi}{2}(X+\uptheta X))\in N_{K}(\mathfrak{{a}})$ and
$\Upomega(\exp(\mathrm{ad}\hskip 0.59998pt\frac{\uppi}{2}(X+\uptheta X)))$ is
the reflection $s_{\upalpha}$ of $\mathfrak{{a}}^{*}$ in the hyperplane
$\upalpha^{\perp}$ (see [7, Prop. 6.52] for a proof). As a consequence, we see
that $\mathrm{W}(\Upsigma)\subseteq\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$. This means that restricted roots lying in
the same orbit of $\mathrm{W}(\Upsigma)$ have the same multiplicities.

We make a choice of positive roots $\Upsigma^{+}\subseteq\Upsigma$ and write
$\Uplambda\subseteq\Upsigma^{+}$ for the subset of simple roots and
$D\subseteq\mathfrak{{a}}^{*}$ for the positive Weyl chamber. The sum
$\mathfrak{{n}}=\bigoplus_{\upalpha\in\Upsigma^{+}}\mathfrak{{g}}_{\upalpha}$
is a nilpotent subalgebra of $\mathfrak{{g}}$, and we have the Iwasawa
decomposition
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{a}}\oplus\mathfrak{{n}}$. Write
$\mathrm{DD}$ for the Dynkin diagram of $\Upsigma$. Just like we did with
$\Upsigma$, we say that $\mathrm{DD}$ is a weighted Dynkin diagram when we
want to stress that each vertex has a positive integer assigned to it, namely
its multiplicity. If a vertex is doubled, i.e. corresponds to a simple root
$\upalpha$ such that $2\upalpha$ is also a root, we assign to it not just one
number, but the ordered pair
$(\mathrm{mult}(\upalpha),\mathrm{mult}(2\upalpha))$.

######  Definition 11.

Let $\mathfrak{{g}}^{\prime}$ be another real semisimple Lie algebra with a
Cartan involution $\uptheta^{\prime}$, a maximal abelian subspace
$\mathfrak{{a}}^{\prime}\subseteq\mathfrak{{p}}^{\prime}$, and a choice of
positive roots $\Upsigma^{\prime+}\subseteq\Upsigma^{\prime}$ fixed. We call a
diagram isomorphism
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$ weighted if it preserves the
vertex weights: $\mathrm{mult}(s(\upalpha))=\mathrm{mult}(\upalpha)$ (and
$\mathrm{mult}(2s(\upalpha))=\mathrm{mult}(2\upalpha)$ in case $2\upalpha$ is
a root) for every $\upalpha\in\Uplambda$. We say that $\mathrm{DD}$ and
$\mathrm{DD}^{\prime}$ are weighted-isomorphic if there exists a weighted
isomorphism between them. Finally, we call a weighted isomorphism
$\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda$
from $\mathrm{DD}$ to itself a weighted automorphism of
$\boldsymbol{\mathrm{DD}}$. The group of all weighted automorphisms of
$\mathrm{DD}$ will be denoted by $\boldsymbol{\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})}\subseteq\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$.

Since $\mathrm{W}(\Upsigma)\subseteq\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$ and $\mathrm{W}(\Upsigma)$ acts
transitively on the set of Weyl chambers in $\mathfrak{{a}}^{*}$, we
immediately get the following:

######  Proposition 12.

Let $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ be real semisimple Lie
algebras with restricted root systems $\Upsigma$ and $\Upsigma^{\prime}$,
respectively. If $\Upsigma$ and $\Upsigma^{\prime}$ are weighted-isomorphic
(in particular, if $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ are
isomorphic), then their Dynkin diagrams are weighted-isomorphic as well for
any choices of $\Upsigma^{+}$ and $\Upsigma^{\prime+}$. In particular, the
Dynkin diagram of $\Upsigma$ does not depend on the choice of $\Upsigma^{+}$
(up to a weighted isomorphism).

Using the results of Section 2, we can prove the converse to Proposition 12.

######  Proposition 13.

Let $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$ be real semisimple Lie
algebras with restricted root systems $\Upsigma$ and $\Upsigma^{\prime}$ and
Dynkin diagrams $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$, respectively. If
$\mathrm{DD}$ and $\mathrm{DD}^{\prime}$ are weighted-isomorphic, then so are
$\Upsigma$ and $\Upsigma^{\prime}$. More specifically, if
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
is a weighted isomorphism between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$,
then its unique linear extension
$s\colon\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$
is a weighted isomorphism between $\Upsigma$ and $\Upsigma^{\prime}$. In
particular, $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})\subseteq\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$.

###### Proof.

We already know from Proposition 6(1) that
$s\colon\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$
is an isomorphism between $\Upsigma$ and $\Upsigma^{\prime}$, so we only need
to prove that it preserves the root multiplicities. We also know from the
proof of Proposition 6 that
$s\mathrm{W}(\Upsigma)s^{-1}=\mathrm{W}(\Upsigma^{\prime})$ and
$\mathrm{W}(\Upsigma)\cdot(\Uplambda\cup(2\Uplambda\cap\Upsigma))=\Upsigma$.
Let $\upalpha\in\Upsigma$ be any root. Take $w\in\mathrm{W}(\Upsigma)$ such
that $w(\upalpha)\in\Uplambda\cup(2\Uplambda\cap\Upsigma)$, and write
$w^{\prime}=sws^{-1}\in\mathrm{W}(\Upsigma^{\prime})$. We have:

$s(\upalpha)=sw^{-1}(w(\upalpha))=w^{\prime-1}s(w(\upalpha)).$

Since $\mathrm{mult}(s(w(\upalpha)))=\mathrm{mult}(w(\upalpha))$ and elements
of the Weyl group preserve root multiplicities, we get
$\mathrm{mult}(s(\upalpha))=\mathrm{mult}(\upalpha)$, so $s$ is a weighted
root system isomorphism. ∎

Since both $\mathrm{W}(\Upsigma)$ and $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$ are contained in $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$ and $\mathrm{W}(\Upsigma)$ acts
transitively on the set of Weyl chambers, we immediately get the following
weighted analog of Proposition 6(3):

######  Corollary 13.1.

$\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)=\mathrm{W}(\Upsigma)\rtimes\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$.

######  Remark 14.

To recapitulate, we know that a weighted-isomorphism class of restricted root
systems yields a weighted-isomorphism class of Dynkin diagrams, and it is
determined by that class. Similarly, an isomorphism class of real semisimple
Lie algebras yields a weighted-isomorphism class of restricted root systems.
We know that it cannot be determined by that class though, since adding a
compact semisimple summand to the Lie algebra would not change the restricted
root system. But it turns out that this is the only obstacle: if
$\mathfrak{{g}}$ has no nonzero compact ideals, it is determined up to
isomorphism by its (weighted) restricted root system – and thus by its
(weighted) Dynkin diagram. The standard proof of this fact, however, is rather
roundabout. One usually first classifies real semisimple Lie algebras –
compact or not – by some other means like Satake or Vogan diagrams, and then
computes explicitly the restricted root system of every Lie algebra in the
classification list. It turns out that non-isomorphic real semisimple Lie
algebras (without nonzero compact ideals) have non-weighted-isomorphic
restricted root systems. The list of the (weighted) restricted root systems of
all noncompact simple Lie algebras can be found in [1, pp. 336-340]. Note that
it is given there in the slightly different – but equivalent – context of
irreducible symmetric spaces of noncompact type (see more on that in Section
4). For example, the restricted root system of
$\mathfrak{{su}}(r,r+n),n\geqslant 1$, is isomorphic to $(BC)_{r}$, and its
Dynkin diagram looks like this:

[arrow style=length=5pt, labels*=2,2,2,2,(2n,1), text style/.style=scale=0.95,
scale=3]B O5

Here we write the simple root multiplicities near the corresponding vertices.
Note that keeping track of the root multiplicities is crucial here: the Lie
algebra $\mathfrak{{sp}}(r,r+n),n\geqslant 1$, also has $(BC)_{r}$ as its
restricted root system, but the multiplicities are different.

Let us look at the homomorphism $\Upomega\colon
N_{K}(\mathfrak{{a}})\to\mathrm{Aut}\hskip 0.59998pt^{\mathrm{{w}}}(\Upsigma)$
through the lens of the semidirect product decomposition $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)=\mathrm{W}(\Upsigma)\rtimes\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$. First off, note that
$\mathrm{Ker}\hskip 0.39993pt(\Upomega)=Z_{K}(\mathfrak{{a}})$. As we have
already seen, $\mathrm{W}(\Upsigma)\subseteq\hskip 0.80002pt\mathrm{Im}\hskip
0.39993pt(\Upomega)$. In fact, for each reflection
$s_{\upalpha}\in\mathrm{W}(\Upsigma)$, we constructed an element in $K^{0}$
that preserves $\mathfrak{{a}}$ and whose image under $\Upomega$ is
$s_{\upalpha}$, which means that
$\mathrm{W}(\Upsigma)\subseteq\Upomega(N_{K^{0}}(\mathfrak{{a}}))$. Here
$N_{K^{0}}(\mathfrak{{a}})$ is a subgroup of $N_{K}(\mathfrak{{a}})$, and it
can also be described as $N_{K}(\mathfrak{{a}})\cap
K^{0}=N_{K}(\mathfrak{{a}})\cap\mathrm{Inn}(\mathfrak{{g}})$ (this equality
follows from the fact that $K$ is a maximal compact subgroup of
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ and thus
$K^{0}=K\cap\mathrm{Aut}\hskip
0.59998pt^{0}(\mathfrak{{g}})=K\cap\mathrm{Inn}(\mathfrak{{g}})$). In
particular, $N_{K^{0}}(\mathfrak{{a}})$ is a normal subgroup of
$N_{K}(\mathfrak{{a}})$. It is proven in [7, Prop. 6.52] that the image of
$N_{K^{0}}(\mathfrak{{a}})$ under $\Upomega$ is precisely
$\mathrm{W}(\Upsigma)$ and thus
$N_{K^{0}}(\mathfrak{{a}})/Z_{K^{0}}(\mathfrak{{a}})\cong\mathrm{W}(\Upsigma)$.

Consider another normalizer subgroup of $K$ given as $N_{K}(\mathfrak{{n}})$.
Let $k\in N_{K}(\mathfrak{{n}})$. As an element of $K$, $k$ commutes with
$\uptheta$ and thus preserves $\uptheta\mathfrak{{n}}$. Since $k$ is
orthogonal with respect to $B_{\uptheta}$, it must preserve
$\mathfrak{{g}}\ominus(\mathfrak{{n}}\oplus\uptheta\mathfrak{{n}})=\mathfrak{{g}}_{0}=\mathfrak{{k}}_{0}\oplus\mathfrak{{a}}$.
But $\mathfrak{{k}}_{0}\subseteq\mathfrak{{k}}$ and
$\mathfrak{{k}}_{0}\perp\mathfrak{{a}}$, so $k$ preserves
$\mathfrak{{k}}_{0}=\mathfrak{{g}}_{0}\cap\mathfrak{{k}}$ and thus
$\mathfrak{{a}}$. We conclude that $N_{K}(\mathfrak{{n}})\subseteq
N_{K}(\mathfrak{{a}})$. Since $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$ consists precisely of those weighted
automorphisms of $\Upsigma$ that preserve the set of positive roots and
$k(\mathfrak{{g}}_{\upalpha})=\mathfrak{{g}}_{\Upomega(k)(\upalpha)}$, we
deduce that $N_{K}(\mathfrak{{n}})=\Upomega^{-1}(\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD}))$.

######  Theorem 15.

Let $\mathfrak{{g}}$ be a real semisimple Lie algebra with
$\uptheta,\mathfrak{{a}},$ and $\Upsigma^{+}$ fixed. Then:

  1. (1)

$\Upomega(N_{K}(\mathfrak{{n}}))=\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$, and hence $\Upomega$ is surjective.

  2. (2)

If $\mathfrak{{g}}$ is simple, then $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})=\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$, and hence $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)=\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$.

Informally, (1) means that every weighted automorphism of $\Upsigma$ can be
lifted to an automorphism of $\mathfrak{{g}}$. Also note that (2) might fail
in case $\mathfrak{{g}}$ is not simple. For instance, if
$\mathfrak{{g}}=\mathfrak{{su}}(r,r+n)\oplus\mathfrak{{sp}}(r,r+n),\,n\geqslant
1,$ then $\Upsigma=(BC)_{r}\sqcup(BC)_{r}$, so $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})=\mathbb{Z}/2\mathbb{Z}$. But the two connected
components of $\mathrm{DD}$ are not weighted-isomorphic, which means that
$\mathrm{Aut}\hskip 0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$ is trivial.

We will prove Theorem 15 by first reducing it to the simple case and then
(mostly) to the theory of complex semisimple Lie algebras.

We need to account for two things: first, $\mathfrak{{g}}$ might have compact
ideals, which make no contribution to the restricted root system, and second,
$\mathfrak{{g}}$ might have isomorphic noncompact simple ideals. To this end,
let us write
$\mathfrak{{g}}=\mathfrak{{g}}_{\mathrm{{c}}}\oplus\mathfrak{{g}}_{1}^{l_{1}}\oplus\cdots\oplus\mathfrak{{g}}_{k}^{l_{k}}$,
where $\mathfrak{{g}}_{\mathrm{{c}}}$ is the sum of all compact ideals of
$\mathfrak{{g}}$, each $\mathfrak{{g}}_{i}$ is a noncompact simple ideal,
$\mathfrak{{g}}_{i}\not\simeq\mathfrak{{g}}_{j}$ for $i\neq j$, and
$\mathfrak{{g}}_{i}^{l_{i}}$ simply stands for
$\bigoplus_{j=1}^{l_{i}}\mathfrak{{g}}_{i}$. Write $l=\sum_{i=1}^{k}l_{i}$ and
let $S_{l}^{l_{1},\ldots,l_{k}}$ stand for the subgroup of the symmetric group
$S_{l}$ consisting of those permutations that permute the first $l_{1}$
elements with each other, the next $l_{2}$ elements with each other, etc.
Clearly, $S_{l}^{l_{1},\ldots,l_{k}}\simeq S_{l_{1}}\times\cdots\times
S_{l_{k}}$. There is an obvious embedding
$S_{l}^{l_{1},\ldots,l_{k}}\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ given by the rule
$\upsigma\cdot(X_{\mathrm{{c}}},(X_{s})_{s=1}^{l})=(X_{\mathrm{{c}}},(X_{\upsigma(s)})_{s=1}^{l})$
(to be precise, this map is an antihomomorphism of groups). We already know
that any Cartan involution on $\mathfrak{{g}}$ respects the decomposition
$\mathfrak{{g}}=\mathfrak{{g}}_{\mathrm{{c}}}\oplus\mathfrak{{g}}_{1}^{l_{1}}\oplus\cdots\oplus\mathfrak{{g}}_{k}^{l_{k}}$.
Without loss of generality, we choose $\uptheta$ so that it is the same on the
isomorphic summands, i.e.
$\uptheta=\left(\mathrm{Id}_{\mathfrak{{g}}_{\mathrm{{c}}}},\prod_{j=1}^{l_{1}}\uptheta_{1},\ldots,\prod_{j=1}^{l_{k}}\uptheta_{k}\right)$.
This way, we have
$\mathfrak{{k}}=\mathfrak{{g}}_{\mathrm{{c}}}\oplus\mathfrak{{k}}_{1}^{l_{1}}\oplus\cdots\oplus\mathfrak{{k}}_{k}^{l_{k}}$
and
$\mathfrak{{p}}=\mathfrak{{p}}_{1}^{l_{1}}\oplus\cdots\oplus\mathfrak{{p}}_{k}^{l_{k}}$.
For such a choice of $\uptheta$, the image of the embedding
$S_{l}^{l_{1},\ldots,l_{k}}\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ actually lies in $N_{K}(\mathfrak{{a}})$. We also
have a subgroup $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{1})^{l_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{k})^{l_{k}}\subseteq\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$. Let $K_{i}=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{i})^{\uptheta_{i}}$ for $1\leqslant i\leqslant k$.
We can choose $\mathfrak{{a}}$ so that
$\mathfrak{{a}}=\mathfrak{{a}}_{1}^{l_{1}}\oplus\cdots\oplus\mathfrak{{a}}_{k}^{l_{k}},\,\mathfrak{{a}}_{i}\subseteq\mathfrak{{p}}_{i}$.
We then have
$\Upsigma=\Upsigma_{1}^{l_{1}}\sqcup\ldots\sqcup\Upsigma_{k}^{l_{k}}$, where
$\Upsigma_{i}\subseteq\mathfrak{{a}}_{i}$ is the restricted root system of
$\mathfrak{{g}}_{i}$ and $\Upsigma_{i}^{l_{i}}$ simply means
$\bigsqcup_{j=1}^{l_{i}}\Upsigma_{i}$. In a similar vein, we have an obvious
embedding $S_{l}^{l_{1},\ldots,l_{k}}\hookrightarrow\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$ and a subgroup $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma_{1})^{l_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma_{k})^{l_{k}}\subseteq\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$. Any choice of positive roots for
$\Upsigma$ is necessarily the union of those for its irreducible components.
We may assume that
$\Upsigma^{+}=(\Upsigma_{1}^{+})^{l_{1}}\sqcup\ldots\sqcup(\Upsigma_{k}^{+})^{l_{k}}$,
hence
$\Uplambda=\Uplambda_{1}^{l_{1}}\sqcup\ldots\sqcup\Uplambda_{k}^{l_{k}}$,
$D=D_{1}^{l_{1}}\times\cdots\times D_{k}^{l_{k}}$, and
$\mathrm{DD}=\mathrm{DD}_{1}^{l_{1}}\sqcup\cdots\sqcup\mathrm{DD}_{k}^{l_{k}}$.

######  Proposition 16.

  1. (1)

The group $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ decomposes as a
semidirect product

$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{1})^{l_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{k})^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

In particular, we have
$\mathrm{Inn}(\mathfrak{{g}})=\mathrm{Inn}(\mathfrak{{g}}_{\mathrm{{c}}})\times\mathrm{Inn}(\mathfrak{{g}}_{1})^{l_{1}}\times\cdots\times\mathrm{Inn}(\mathfrak{{g}}_{k})^{l_{k}}$.

  2. (2)

The group $K$ decomposes as a semidirect product

$K=\left[\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times
K_{1}^{l_{1}}\times\cdots\times K_{k}^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

In particular, we have
$K^{0}=\mathrm{Inn}(\mathfrak{{g}}_{\mathrm{{c}}})\times(K_{1}^{0})^{l_{1}}\times\cdots\times(K_{k}^{0})^{l_{k}}$.

  3. (3)

The group $N_{K}(\mathfrak{{a}})$ decomposes as a semidirect product

$N_{K}(\mathfrak{{a}})=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times
N_{K_{1}}(\mathfrak{{a}}_{1})^{l_{1}}\times\cdots\times
N_{K_{k}}(\mathfrak{{a}}_{k})^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

In particular, we have
$N_{K^{0}}(\mathfrak{{a}})=\mathrm{Inn}(\mathfrak{{g}}_{\mathrm{{c}}})\times
N_{K_{1}^{0}}(\mathfrak{{a}}_{1})^{l_{1}}\times\cdots\times
N_{K_{k}^{0}}(\mathfrak{{a}}_{k})^{l_{k}}$.

  4. (4)

The group $N_{K}(\mathfrak{{n}})$ decomposes as a semidirect product

$N_{K}(\mathfrak{{n}})=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times
N_{K_{1}}(\mathfrak{{n}}_{1})^{l_{1}}\times\cdots\times
N_{K_{k}}(\mathfrak{{n}}_{k})^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

  5. (5)

The group $\mathrm{Aut}\hskip 0.59998pt^{\mathrm{{w}}}(\Upsigma)$ decomposes
as a semidirect product

$\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)=\left[\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma_{1})^{l_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma_{k})^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

  6. (6)

The group $\mathrm{W}(\Upsigma)$ decomposes as a product

$\mathrm{W}(\Upsigma)=\mathrm{W}(\Upsigma_{1})^{l_{1}}\times\cdots\times\mathrm{W}(\Upsigma_{k})^{l_{k}}.$

  7. (7)

The group $\mathrm{Aut}\hskip 0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$
decomposes as a semidirect product

$\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})=\left[\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD}_{1})^{l_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD}_{k})^{l_{k}}\right]\rtimes
S_{l}^{l_{1},\ldots,l_{k}}.$

  8. (8)

With respect to the decompositions (3) and (5), the homomorphism
$\Upomega\colon N_{K}(\mathfrak{{a}})\to\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$ decomposes as

$\Upomega=\left(E,\Upomega_{1}^{l_{1}},\ldots,\Upomega_{k}^{l_{k}},\mathrm{Id}_{S_{l}^{l_{1},\ldots,l_{k}}}\right),$

where $E$ is the trivial homomorphism $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\to\left\\{\hskip 0.29999pt{e}\hskip
0.29999pt\right\\}$, $\Upomega_{i}\colon
N_{K_{i}}(\mathfrak{{a}}_{i})\to\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma_{i})$, and the last component
$\mathrm{Id}_{S_{l}^{l_{1},\ldots,l_{k}}}$ formally means that the following
diagram commutes:

---
$\textstyle{N_{K}(\mathfrak{{a}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle\Upomega}$$\textstyle{S_{l}^{l_{1},\ldots,l_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)}$

We omit the proof, as it simply boils down to the fact that any automorphism
of $\mathfrak{{g}}$ must preserve $\mathfrak{{g}}_{\mathrm{{c}}}$ and permute
the remaining noncompact simple ideals, and the same is true for weighted
automorphisms of
$\Upsigma=\Upsigma_{1}^{l_{1}}\sqcup\ldots\sqcup\Upsigma_{k}^{l_{k}}$.

Part (8) of Proposition 16 implies that if
$\Upomega_{i}(N_{K_{i}}(\mathfrak{{n}}_{i}))=\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD}_{i})$ for each $i$, then
$\Upomega(N_{K}(\mathfrak{{n}}))=\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$. Consequently, in order to prove
Theorem 15, we may restrict to the case when $\mathfrak{{g}}$ is simple and
noncompact. We will actually show that
$\Upomega(N_{K}(\mathfrak{{n}}))=\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})$ in
this case, thus proving both parts (1) and (2) of the theorem.

We will consider three different scenarios. To begin with, we can immediately
cast aside all those simple Lie algebras where $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ (and hence $\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$) is trivial. This leaves us with those
Lie algebras where $\Upsigma=A_{n}\,(n\geqslant 2),D_{n}\,(n\geqslant 4),$ or
$E_{6}$.

Each complex semisimple Lie algebra $\mathfrak{{g}}$ gives rise to at least
two noncompact real ones: the realification and the split real form of
$\mathfrak{{g}}$. These are going to be our first two scenarios. As a matter
fact, here we do not require the Lie algebra to be simple, and we will only
use that assumption in the third scenario. Let $\mathfrak{{g}}$ be a complex
semisimple Lie algebra, and let $\mathfrak{{h}},\Updelta,\Uplambda,$ and
$\left\\{\hskip 0.29999pt{h_{i},e_{i},f_{i}}\hskip
0.29999pt\right\\}_{i=1}^{r}$ be as on page 6. It follows from the Isomorphism
Theorem that there exists a unique automorphism $\uptheta$ of $\mathfrak{{g}}$
as of a real Lie algebra that is $\mathbb{C}$-antilinear and satisfies

$\uptheta(h_{i})=-h_{i},\quad\uptheta(e_{i})=-f_{i},\quad\uptheta(f_{i})=-e_{i}.$
(1)

This automorphism is involutive and is in fact a compact real structure and a
Cartan involution555The latter two notions coincide for any complex semisimple
Lie algebra, see [10, Sec. 5] (hence the notation). In particular,
$\mathfrak{{p}}=i\mathfrak{{k}}$. Every Cartan involution on $\mathfrak{{g}}$
is of this form (for some choice of $\mathfrak{{h}},\Uplambda,$ and canonical
generators). We can introduce two more involutive automorphisms of
$\mathfrak{{g}}$: the Weyl involution $\upomega$ and the split real structure
$\uptau$. The Weyl involution is given on the canonical generators by the same
formula (1) but is $\mathbb{C}$-linear, whereas the split real form fixes all
the canonical generators but is $\mathbb{C}$-antilinear. Once again, the
existence and uniqueness of both of these automorphisms follow from the
Isomorphism Theorem. Clearly, the three automorphisms commute pairwise and the
product of any two of them equals the third one. The fixed point (real)
subalgebra $\mathfrak{{g}}^{\uptau}$ is the split real form of
$\mathfrak{{g}}$, and every split real form is of this form (for some choice
of $\mathfrak{{h}},\Uplambda,$ and canonical generators).

Scenario 1: the realification. Here we assume that our real semisimple Lie
algebra is the realification of a complex one and use the notation established
in the previous paragraph. In particular, the Cartan involution $\uptheta$ is
given on the canonical generators by (1). Write
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$ for the corresponding
Cartan decomposition. Since $\uptheta$ is $\mathbb{C}$-antilinear and
$\uptheta(h_{i})=-h_{i}$ for each $i$, we have
$\mathfrak{{h}}\cap\mathfrak{{p}}=\mathfrak{{h}}(\mathbb{R}),\,\mathfrak{{h}}\cap\mathfrak{{k}}=i\mathfrak{{h}}(\mathbb{R})$.
We claim that $\mathfrak{{h}}(\mathbb{R})$ is a maximal abelian subspace of
$\mathfrak{{p}}$. Indeed, let $\mathfrak{{b}}\subset\mathfrak{{p}}$ be an
abelian subspace containing $\mathfrak{{h}}(\mathbb{R})$. If we think of
$\mathfrak{{g}}$ as a real Euclidean vector space (with respect to the inner
product $B_{\uptheta}$), then all operators of the form $\mathrm{ad}\hskip
0.59998pt(X),\,X\in\mathfrak{{p}},$ are self-adjoint (this is true for any
real semisimple Lie algebra). Since such operators are also
$\mathbb{C}$-linear, they are also self-adjoint with respect to the Hermitian
inner product $\langle\hskip 1.00006pt{X}\hskip 1.00006pt|\hskip
1.00006pt{Y}\hskip 1.00006pt\rangle\hskip
0.20004pt=B_{\uptheta}(X,Y)+iB_{\uptheta}(X,iY)$ and hence diagonalizable over
$\mathbb{C}$. It follows that
$\mathfrak{{b}}_{\mathbb{C}}=i\mathfrak{{b}}\oplus\mathfrak{{b}}$ (here
$i\mathfrak{{b}}\subseteq i\mathfrak{{p}}=\mathfrak{{k}}$) is an abelian
complex subalgebra of $\mathfrak{{g}}$ consisting of semisimple elements and
containing $\mathfrak{{h}}$, hence we must have
$\mathfrak{{b}}_{\mathbb{C}}=\mathfrak{{h}}$ and thus
$\mathfrak{{b}}=\mathfrak{{h}}(\mathbb{R})$. So we can write
$\mathfrak{{a}}=\mathfrak{{h}}(\mathbb{R})$. In this case,
$\mathfrak{{a}}^{*}=\mathfrak{{h}}^{*}(\mathbb{R})$, and the root system
$\Updelta$ of $\mathfrak{{g}}$ as of a complex semisimple Lie algebra
coincides with the restricted root system $\Upsigma$ of $\mathfrak{{g}}$ as of
a real semisimple Lie algebra (for our specific choice of $\uptheta$ and
$\mathfrak{{a}}$). Moreover, the root space decomposition and the restricted
root space decomposition coincide as well. Note that
$\mathfrak{{g}}_{0}=\mathfrak{{h}}=i\mathfrak{{a}}\oplus\mathfrak{{a}}$ and
$\mathfrak{{k}}_{0}=i\mathfrak{{a}}$. Each restricted root subspace
$\mathfrak{{g}}_{\upalpha},\upalpha\in\Upsigma,$ thus has real dimension 2,
i.e. all of the root multiplicities equal 2. As we know from Section 2, every
(not necessarily weighted) automorphism $s\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ can be lifted to a (complex) automorphism $\hat{s}$ of
$\mathfrak{{g}}$ given by the rule
$\hat{s}(h_{i})=h_{s(i)},\hat{s}(e_{i})=e_{s(i)},\hat{s}(f_{i})=f_{s(i)}$.
This automorphism satisfies $\hat{s}(\mathfrak{{a}})=\mathfrak{{a}}$ and
$\Uppsi(\hat{s})=({\left.\kern-1.2pt\hat{s}\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}^{*})^{-1}=s$.
It is easily seen on the canonical generators that $\hat{s}$ commutes with
$\uptheta$ and preserves $\mathfrak{{n}}$. Consequently, $\hat{s}\in
N_{K}(\mathfrak{{n}})$ and $\Upomega(\hat{s})=\Uppsi(\hat{s})=s$, which
finishes the proof in this scenario.

Scenario 2: split real form. Now suppose that our real semisimple Lie algebra
is a split real form of its complexification. We denote the latter by
$\mathfrak{{g}}$ and use the same notation as above. Then our real semisimple
Lie algebra can be described as $\mathfrak{{g}}^{\uptau}$. One can easily see
that $\mathfrak{{g}}^{\uptau}$ is the real Lie subalgebra of $\mathfrak{{g}}$
generated by $e_{i},f_{i},$ and $h_{i},1\leqslant i\leqslant r$. The
automorphism $\uptheta$ of $\mathfrak{{g}}$ clearly preserves
$\mathfrak{{g}}^{\uptau}$. Since $\mathfrak{{g}}^{\uptau}$ is a real form of
$\mathfrak{{g}}$, the Killing form $B$ of $\mathfrak{{g}}$ is the
$\mathbb{C}$-bilinear extension of the Killing form $B^{\uptau}$ of
$\mathfrak{{g}}^{\uptau}$. This implies that the restriction of $\uptheta$ to
$\mathfrak{{g}}^{\uptau}$ is a Cartan involution on $\mathfrak{{g}}^{\uptau}$.
Moreover, if we write the corresponding Cartan decomposition as
$\mathfrak{{g}}^{\uptau}=\mathfrak{{k}}\oplus\mathfrak{{p}}$, then
$\mathfrak{{h}}(\mathbb{R})$ lies in $\mathfrak{{p}}$ and is its maximal
abelian subspace. Indeed, for each $X\in\mathfrak{{p}}$, the operator
$\mathrm{ad}\hskip 0.59998pt_{\mathfrak{{g}}^{\uptau}}(X)$ is diagonalizable
over $\mathbb{R}$, hence its $\mathbb{C}$-linear extension $\mathrm{ad}\hskip
0.59998pt_{\mathfrak{{g}}}(X)$ is diagonalizable over $\mathbb{C}$ as an
operator on $\mathfrak{{g}}$. Therefore, in the same fashion as above, the
existence of a larger abelian subspace of $\mathfrak{{p}}$ would lead to a
toral subalgebra of $\mathfrak{{g}}$ larger than $\mathfrak{{h}}$, hence a
contradiction. Once again, we can take
$\mathfrak{{a}}=\mathfrak{{h}}(\mathbb{R})$, in which case
$\mathfrak{{a}}^{*}=\mathfrak{{h}}^{*}(\mathbb{R})$ and the restricted root
system $\Upsigma$ of $\mathfrak{{g}}^{\uptau}$ coincides with $\Updelta$. Just
as before, every diagram automorphism $s\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ can be lifted to the complex automorphism $\hat{s}\in
N_{\mathrm{Aut}\hskip 0.41998pt(\mathfrak{{g}})}(\mathfrak{{h}})$ of
$\mathfrak{{g}}$ such that $\Uppsi(\hat{s})=s$, and it can be seen from the
defining formula for $\hat{s}$ that it commutes with both $\uptheta$ and
$\uptau$. In particular, it preserves $\mathfrak{{g}}^{\uptau}$ and the
restriction
${\left.\kern-1.2pt\hat{s}\vphantom{\big{|}}\right|_{\mathfrak{{g}}^{\uptau}}}$
lies in $N_{K}(\mathfrak{{n}})$. We have
$\Upomega({\left.\kern-1.2pt\hat{s}\vphantom{\big{|}}\right|_{\mathfrak{{g}}^{\uptau}}})=\Uppsi(\hat{s})=s$,
which finishes the proof in scenario 2.

Scenario 3: the rest. Now we go back to our assumption that $\mathfrak{{g}}$
is simple and $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})$ is nontrivial. An
examination of the list of all real simple noncompact Lie algebras ([1, pp.
336-340]) reveals that if $\mathfrak{{g}}$ is neither split nor complex, it
has to be isomorphic to either $\mathfrak{{sl}}(n,\mathbb{H})\,(n\geqslant 3)$
or $\mathfrak{{e}}_{6}^{-26}$. The restricted root systems of these Lie
algebras are $A_{n-1}$ and $A_{2}$, respectively. In both cases,
$\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})\cong\mathbb{Z}/2\mathbb{Z}$, and
there is only one nontrivial diagram automorphism that we want to lift to
$N_{K}(\mathfrak{{n}})$. Recall that we have a distinguished automorphism
$\uptheta$ of $\mathfrak{{g}}$ fixed. Plainly, $\uptheta\in
N_{K}(\mathfrak{{a}})$ and
$\Upomega(\uptheta)=-\mathrm{Id}_{\mathfrak{{a}}^{*}}$. The weighted root
system automorphism $-\mathrm{Id}_{\mathfrak{{a}}^{*}}$ can be decomposed as
$-\mathrm{Id}_{\mathfrak{{a}}^{*}}=w_{0}s$, where
$w_{0}\in\mathrm{W}(\Upsigma)$ and $s\in\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$. Here $s(D)=D$ and
$-\mathrm{Id}_{\mathfrak{{a}}^{*}}(D)=-D$, so $w_{0}(D)=-D$ (this uniquely
determines $w_{0}$ and also shows that it is the longest element of
$\mathrm{W}(\Upsigma)$ with respect to the system of generators
$s_{\upalpha_{1}},\ldots,s_{\upalpha_{r}}$). The diagram automorphism
$s=-w_{0}$ may or may not be trivial, depending on $\Upsigma$. Note that this
construction does not really rely on $\mathfrak{{g}}$, nor does it use root
multiplicities, so it can be carried out for any root system $(V,\Updelta)$:
pick $\Updelta^{+}$ and decompose $-\mathrm{Id}_{V}\in\mathrm{Aut}\hskip
0.59998pt(\Updelta)$ as $-\mathrm{Id}_{V}=w_{0}s$ with respect to the
semidirect product decomposition $\mathrm{Aut}\hskip
0.59998pt(\Updelta)=\mathrm{W}(\Updelta)\rtimes\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$. It was shown in [10, §4, Prop. 4] that, in case
$\Updelta$ is irreducible, $s$ is a nontrivial diagram automorphism precisely
when $\Updelta=A_{n}\,(n\geqslant 2),D_{2n+1}(n\geqslant 2),$ or $E_{6}$. This
covers both of our cases
$\mathfrak{{g}}=\mathfrak{{sl}}(n,\mathbb{H})\,(n\geqslant 3)$ and
$\mathfrak{{g}}=\mathfrak{{e}}_{6}^{-26}$. Now, we know from the discussion
after Proposition 10 that there exists $\upvarphi\in N_{K}(\mathfrak{{a}})$
such that $\Upomega(\upvarphi)=w_{0}$. We have $\upvarphi\uptheta\in
N_{K}(\mathfrak{{n}})$ and $\Upomega(\upvarphi\uptheta)=w_{0}^{2}s=s$. In
other words, the only nontrivial element of $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD})$ lies in the image of $\Upomega$ and so
$\Upomega(N_{K}(\mathfrak{{n}}))=\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD})$,
which completes the proof of Theorem 15.

######  Remark 17.

Part (2) of Theorem 15 can also be proven simply by examining the
classification of simple noncompact Lie algebras and the list of their
(weighted) Dynkin diagrams ([1, pp. 336-340]). Indeed, for any such diagram,
if there are two vertices that differ by a (not necessarily weighted) diagram
automorphism, then they happen to have the same multiplicity, which implies
that every diagram automorphism is weighted.

######  Corollary 15.1.

Let $\mathfrak{{g}}$ be a real semisimple Lie algebra. Then we have
$N_{K}(\mathfrak{{a}})/Z_{K}(\mathfrak{{a}})\cong\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)$ and
$N_{K}(\mathfrak{{n}})/Z_{K}(\mathfrak{{n}})\cong\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\mathrm{DD})$.

######  Corollary 15.2.

Let $\mathfrak{{g}},\mathfrak{{g}}^{\prime}$ be real semisimple Lie algebras
with
$\uptheta,\uptheta^{\prime},\mathfrak{{a}},\mathfrak{{a}}^{\prime},\Upsigma^{+},$
and $\Upsigma^{\prime+}$ fixed.

  1. (1)

Every weighted isomorphism
$f\colon\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$
between $\Upsigma$ and $\Upsigma^{\prime}$ is an isometry. In particular,
$\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)\subseteq\mathrm{O}(\mathfrak{{a}}^{*})$.

  2. (2)

Now assume that neither $\mathfrak{{g}}$ nor $\mathfrak{{g}}^{\prime}$ have
nonzero compact ideals. Then for every weighted isomorphism
$f\colon\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime*}$
between $\Upsigma$ and $\Upsigma^{\prime}$, there exists a Lie algebra
isomorphism
$F\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
such that $F\circ\uptheta=\uptheta^{\prime}\circ F$ and
$F(\mathfrak{{a}})=\mathfrak{{a}}^{\prime}$ and the induced weighted
isomorphism
$\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime*}$
between $\Upsigma$ and $\Upsigma^{\prime}$ coincides with $f$. In particular,
for every weighted diagram isomorphism
$s\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
between $\mathrm{DD}$ and $\mathrm{DD}^{\prime}$, there exists such
$F\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$
that the induced diagram isomorphism
${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{\Uplambda}}\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda^{\prime}$
coincides with $s$.

###### Proof.

For part (2), we know from Remark 14 that there exists some Lie algebra
isomorphism
$\widetilde{{F}}\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}$.
As explained in the proof of Proposition 10, we may assume
$\widetilde{{F}}\circ\uptheta=\uptheta^{\prime}\circ\widetilde{{F}}$ and
$\widetilde{{F}}(\mathfrak{{a}})=\mathfrak{{a}}^{\prime}$. Let
$\widetilde{{f}}\colon\mathfrak{{a}}^{*}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{\prime*}$
be the induced weighted isomorphism between $\Upsigma$ and
$\Upsigma^{\prime}$. Then $f\circ\widetilde{{f}}^{-1}\in\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma^{\prime})$. According to Theorem 15, there
exists $\upvarphi^{\prime}\in N_{K^{\prime}}(\mathfrak{{a}}^{\prime})$ such
that $\Upomega^{\prime}(\upvarphi^{\prime})=f\circ\widetilde{{f}}^{-1}$. Then
the weighted root system isomorphism between $\Upsigma$ and
$\Upsigma^{\prime}$ induced by $F=\upvarphi^{\prime}\circ\widetilde{{F}}$ is
$\Upomega^{\prime}(\upvarphi^{\prime})\circ\widetilde{{f}}=f$. For part (1),
write
$\mathfrak{{g}}=\mathfrak{{g}}_{\mathrm{{c}}}\oplus\mathfrak{{g}}_{\mathrm{{nc}}}$,
where $\mathfrak{{g}}_{\mathrm{{c}}}$ is the maximal compact ideal and
$\mathfrak{{g}}_{\mathrm{{nc}}}$ is the complementary noncompact ideal, and do
the same for $\mathfrak{{g}}^{\prime}$:
$\mathfrak{{g}}^{\prime}=\mathfrak{{g}}^{\prime}_{\mathrm{{c}}}\oplus\mathfrak{{g}}^{\prime}_{\mathrm{{nc}}}$.
The restricted root systems of $\mathfrak{{g}}$ and $\mathfrak{{g}}^{\prime}$
coincide with those of $\mathfrak{{g}}_{\mathrm{{nc}}}$ and
$\mathfrak{{g}}^{\prime}_{\mathrm{{nc}}}$, respectively, so every weighted
isomorphism between them is an isometry by part (2) and our observation after
Definition 9. ∎

##  4\. Symmetric spaces of noncompact type

The results of the previous section have a useful interpretation in terms of
the theory of symmetric spaces of noncompact type, morally because those are
’the same’ as noncompact real semisimple Lie algebras. See [6, Ch. IV-VI] for
a background on symmetric spaces.

Let $M$ be a symmetric space of noncompact type. It is well known that the
isometry group $I(M)$ is a Lie group in the compact-open topology and its
action on $M$ is smooth. We denote it by $\widetilde{G}$ and its Lie algebra
by $\mathfrak{{g}}$. We have a map $\mathfrak{{g}}\to\mathfrak{{X}}(M)$,
called the infinitesimal generator of the (transitive) action
$\widetilde{{G}}\curvearrowright M$, that sends $X$ to the corresponding
fundamental vector field $X^{*}$. This map is an injective anti-homomorphism
of Lie algebras and its image is precisely the Lie subalgebra
$\mathcal{K}(M)\subseteq\mathfrak{{X}}(M)$ of Killing vector fields. Denote
$G=\widetilde{G}^{0}$. Since $M$ is connected, the action $G\curvearrowright
M$ is transitive as well.

Pick any $o\in M$, write $\widetilde{K}$ for the isotropy subgroup of
$\widetilde{{G}}$ at $o$, and denote $K=\widetilde{{K}}\cap G$. The action
$\widetilde{{G}}\curvearrowright M$ (as well as $G\curvearrowright M$) is
proper (see [3]), which implies that $\widetilde{{K}}$ is a compact subgroup
of $\widetilde{{G}}$. Moreover, $\widetilde{{G}}/\widetilde{{K}}=M$ is
contractible, so $\widetilde{{K}}$ is a maximal compact subgroup of
$\widetilde{{G}}$, $K$ is a maximal compact subgroup of $G$, and
$K=\widetilde{{K}}^{0}$ is connected. We have the geodesic symmetry
$s_{o}\in\widetilde{{G}}$ at $o$, which gives rise to an involutive
automorphism $\Uptheta=C_{s_{o}}\colon g\mapsto s_{o}gs_{o}^{-1}=s_{o}gs_{o}$
of $\widetilde{{G}}$ and thus to an involutive automorphism
$\uptheta=(C_{s_{o}})_{*}=\mathrm{Ad}\hskip 0.59998pt(s_{o})$ of
$\mathfrak{{g}}$. Since $M$ is of noncompact type, $\widetilde{{G}}$ (and thus
$\mathfrak{{g}}$) is semisimple and $\uptheta$ is a Cartan involution on
$\mathfrak{{g}}$, so we can write
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$ for the corresponding
Cartan decomposition666In this section, we are no longer using the letter $K$
to denote $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{\uptheta}$.. One can
show that $\widetilde{{K}}$ is an open subgroup of
$\widetilde{{G}}^{\Uptheta}$, which means that $\mathrm{Lie}\hskip
0.59998pt(\widetilde{{K}})=\mathfrak{{k}}$ and $\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}(\widetilde{{K}})\subseteq\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$. In particular, the adjoint
representation of $\widetilde{{K}}$ on $\mathfrak{{g}}$ is orthogonal with
respect to $B_{\uptheta}$ and preserves $\mathfrak{{k}}$ and $\mathfrak{{p}}$.

Consider the orbit map $\widetilde{{G}}\twoheadrightarrow M,g\mapsto g\cdot
o$. Its differential at $e$ is a surjective linear map
$\uppi\colon\mathfrak{{g}}\twoheadrightarrow T_{o}M$ with $\mathrm{Ker}\hskip
0.39993pt(\uppi)=\mathfrak{{k}}$, so it maps $\mathfrak{{p}}$ isomorphically
onto $T_{o}M$. We always identify $\mathfrak{{p}}$ and $T_{o}M$ by means of
this isomorphism. One can easily show that this is an isomorphism of
$\widetilde{{K}}$-representations: between the adjoint representation of
$\widetilde{{K}}$ in $\mathfrak{{p}}$ and its isotropy representation in
$T_{o}M$.

######  Remark 18.

Note that if we were to pick another point $o^{\prime}=g\cdot o\in M$, we
would have $s_{o^{\prime}}=gs_{o}g^{-1}$ and hence
$\uptheta^{\prime}=\mathrm{Ad}\hskip
0.59998pt(s_{o^{\prime}})=\mathrm{Ad}\hskip
0.59998pt(g)\uptheta\mathrm{Ad}\hskip 0.59998pt(g)^{-1}$. Consequently, the
corresponding Cartan decompositions differ by $\mathrm{Ad}\hskip
0.59998pt(g)$: $\mathrm{Ad}\hskip
0.59998pt(g)(\mathfrak{{k}})=\mathfrak{{k}}^{\prime},\mathrm{Ad}\hskip
0.59998pt(g)(\mathfrak{{p}})=\mathfrak{{p}}^{\prime}$.

If we denote the Riemannian metric on $M$ by $g$, then $g_{o}$ is a
$\widetilde{{K}}$-invariant inner product on $T_{o}M$, which is the same as an
$\mathrm{Ad}\hskip 0.59998pt_{\widetilde{{G}}}(\widetilde{{K}})$-invariant
inner product on $\mathfrak{{p}}$. From the previous section, we know that
another such inner product is
${\left.\kern-1.2ptB_{\uptheta}\vphantom{\big{|}}\right|_{\mathfrak{{p}}\times\mathfrak{{p}}}}={\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}\times\mathfrak{{p}}}}$.
In general, these two inner products may not coincide. However, the difference
between them allows an explicit description.

Let $M=M_{1}\times\cdots\times M_{k}$ be the de Rham decomposition of $M$ (see
[8, Ch. IV, Sec. 5,6] for more on the de Rham decomposition). We say that
$M_{i}$ and $M_{j}$ are homothetic and write $\boldsymbol{M_{i}\sim M_{j}}$ if
there exists a diffeomorphism $\upvarphi\colon
M_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}M_{j}$
such that $\upvarphi^{*}g_{j}=ag_{i}$ for some $a>0$ (i.e., $\upvarphi$ is
’almost’ an isometry: it is conformal with a constant conformal factor). This
is an equivalence relation on the set of de Rham factors, and it is in general
weaker than $M_{i}\simeq M_{j}$ (being isometric). The isometry group of $M$
allows a description similar to that of $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ (established in Proposition 16(1)). We will present
it in a slightly different form without grouping isometric de Rham factors
together. We define:

$\displaystyle\boldsymbol{S_{k}^{\simeq}}=\left\\{\hskip 0.29999pt{\upsigma\in
S_{k}\mid M_{i}\simeq M_{\upsigma(i)}\;\forall\,i\in\left\\{\hskip
0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}}\hskip 0.29999pt\right\\},$
$\displaystyle\boldsymbol{S_{k}^{\sim}}=\left\\{\hskip 0.29999pt{\upsigma\in
S_{k}\mid M_{i}\sim M_{\upsigma(i)}\;\forall\,i\in\left\\{\hskip
0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}}\hskip 0.29999pt\right\\}.$

These are subgroups of the symmetric group $S_{k}$, and we have
$S_{k}^{\simeq}\subseteq S_{k}^{\sim}$. Note that these definitions only make
sense after we fix the (order in the) de Rham decomposition of $M$. For any
pair of indices $i,j\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\}$ such that $M_{i}\simeq M_{j}$, pick an isometry
$\upvarphi_{ij}\colon
M_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}M_{j}$
in such a way that if we have $M_{i}\simeq M_{j}\simeq M_{l}$, then
$\upvarphi_{jl}\circ\upvarphi_{ij}=\upvarphi_{il}$. This gives an embedding

$\displaystyle S_{k}^{\simeq}\hookrightarrow
I(M),\upsigma\mapsto\upvarphi_{\upsigma},\;\text{where}$
$\displaystyle\upvarphi_{\upsigma}(p_{1},\ldots,p_{k})=(\upvarphi_{\upsigma(1)1}(p_{\upsigma(1)}),\ldots,\upvarphi_{\upsigma(k)k}(p_{\upsigma(k)}))$

(to be precise, this is an injective group anti-homomorphism). Surely, this
embedding depends on the choice of $\upvarphi_{ij}$’s. We also have an obvious
embedding $I(M_{1})\times\cdots\times I(M_{k})\subseteq I(M)$. The following
result is proven in [11] (in much greater generality and in a slightly
different notation more akin to the one we used in Proposition 16):

######  Fact 19.

Let $M$ be a symmetric space of noncompact type and $M=M_{1}\times\dots\times
M_{k}$ its de Rham decomposition. Then the group $I(M)$ decomposes as a
semidirect product

$I(M)=\left[I(M_{1})\times\cdots\times I(M_{k})\right]\rtimes S_{k}^{\simeq}.$

In particular, we have $I^{0}(M)=I^{0}(M_{1})\times\cdots\times I^{0}(M_{k})$.

More generally, if $M^{\prime}$ is another symmetric space of noncompact type
with the de Rham decomposition $M^{\prime}=M^{\prime}_{1}\times\cdots\times
M^{\prime}_{k}$ and $\upvarphi\colon
M\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}M^{\prime}$
is an isometry, then there exists a permutation $\upsigma\in S_{k}$ and a
collection of isometries $\upvarphi_{i}\colon
M_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}M^{\prime}_{\upsigma^{-1}(i)},1\leqslant
i\leqslant k$, such that

$\upvarphi(p_{1},\ldots,p_{k})=(\upvarphi_{\upsigma(1)}(p_{\upsigma(1)}),\ldots,\upvarphi_{\upsigma(k)}(p_{\upsigma(k)})).$

As a consequence, the de Rham decomposition of $M$ is essentially unique up to
reordering of the factors.

Let us write
$\widetilde{{G}}_{i}=I(M_{i}),G_{i}=\widetilde{{G}}_{i}^{0},\mathfrak{{g}}_{i}=\mathrm{Lie}\hskip
0.59998pt(\widetilde{{G_{i}}})$. Denote $o=(o_{1},\ldots,o_{k})$ and write
$\widetilde{{K}}_{i}$ for the stabilizer subgroup of $\widetilde{{G}}_{i}$ at
$o_{i}$ and $K_{i}=\widetilde{{K}}_{i}\cap G_{i}=\widetilde{{K}}_{i}^{0}$. We
necessarily have $s_{o}=(s_{o_{1}},\ldots s_{o_{k}})$,
$\Uptheta=(\Uptheta_{1},\ldots,\Uptheta_{k})$, and
$\uptheta=(\uptheta_{1},\ldots,\uptheta_{k})$, where
$\uptheta_{i}=(\Uptheta_{i})_{*}=\mathrm{Ad}\hskip 0.59998pt(s_{o_{i}})$. The
latter decomposition of $\uptheta$ implies that we have
$\mathfrak{{k}}=\bigoplus_{i=1}^{k}\mathfrak{{k}}_{i}$ and
$\mathfrak{{p}}=\bigoplus_{i=1}^{k}\mathfrak{{p}}_{i}$, just like we had on
page 15. It is well know from the theory of symmetric spaces that $M$ is (de
Rham) irreducible if and only if its isometry Lie algebra $\mathfrak{{g}}$ is
simple (this is no longer true for symmetric spaces of compact type). In
particular, each $\mathfrak{{g}}_{i}$ is simple (and noncompact). We stress
that $\mathfrak{{g}}$ has no nonzero compact ideals. The image of the
restricted isotropy representation
$K_{i}\hookrightarrow\mathrm{O}(T_{o_{i}}M_{i})$ coincides with the restricted
holonomy group $\mathrm{Hol}^{0}(M_{i},o_{i})$ (this is true for all
semisimple symmetric spaces). In particular, this representation is
irreducible and thus any two $K_{i}$-invariant inner products on
$\mathfrak{{p}}_{i}\cong T_{o_{i}}M_{i}$ are proportional. We already have two
such inner products:
${\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}\times\mathfrak{{p}}_{i}}}$
and $(g_{i})_{o_{i}}$ (here $g_{i}$ is the Riemannian metric on $M_{i}$).
Therefore, we have
$(g_{i})_{o_{i}}=\uplambda_{i}{\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}\times\mathfrak{{p}}_{i}}}$,
and we can write
$g_{o}=\uplambda_{1}{\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{1}\times\mathfrak{{p}}_{1}}}+\cdots+\uplambda_{k}{\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{k}\times\mathfrak{{p}}_{k}}}$.
We call $\uplambda_{1},\ldots,\uplambda_{k}$ the normalizing constants of
$\boldsymbol{M}$.

######  Lemma 20.

The normalizing constants of $M$ are well defined up to reordering.

###### Proof.

Formally, the lemma means that if we are given another de Rham decomposition
$M=M^{\prime}_{1}\times\cdots\times M^{\prime}_{k}$ and base point
$o^{\prime}=(o^{\prime}_{1},\ldots,o^{\prime}_{k})$ with the corresponding
normalizing constants $\uplambda^{\prime}_{1},\ldots,\uplambda^{\prime}_{k}$,
then there exists a permutation $\upsigma\in S_{k}$ such that $M_{i}\simeq
M^{\prime}_{\upsigma(i)}$ and $\uplambda_{i}=\uplambda^{\prime}_{\upsigma(i)}$
for each $i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$.
According to Fact 19, there exists $\upsigma\in S_{k}$ satisfying the former
condition: $M_{i}\simeq M^{\prime}_{\upsigma(i)}$. Since each de Rham factor
is a symmetric space in its own right, it is in particular a Riemannian
homogeneous space, so we can pick isometries $\upvarphi_{i}\colon
M_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}M^{\prime}_{\upsigma(i)},1\leqslant
i\leqslant k$, such that $\upvarphi_{i}(o_{i})=o^{\prime}_{\upsigma(i)}$. We
have a Lie group isomorphism
$F_{i}\colon\widetilde{{G}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\widetilde{{G}}^{\prime}_{\upsigma(i)},g\mapsto\upvarphi_{i}\circ
g\circ\upvarphi_{i}^{-1},$ such that
$F_{i}(s_{o_{i}})=s_{o^{\prime}_{\upsigma(i)}}$, and thus the induced Lie
algebra isomorphism
$f_{i}\colon\mathfrak{{g}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}^{\prime}_{\upsigma(i)}$
satisfies $f_{i}(\mathfrak{{k}}_{i})=\mathfrak{{k}}^{\prime}_{\upsigma(i)}$
and $f_{i}(\mathfrak{{p}}_{i})=\mathfrak{{p}}^{\prime}_{\upsigma(i)}$.
Plainly,
$f_{i}\colon\mathfrak{{p}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{p}}^{\prime}_{\upsigma(i)}$
is an isometry with respect to the inner products
${\left.\kern-1.2ptB_{i}\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}\times\mathfrak{{p}}_{i}}}$
and
${\left.\kern-1.2ptB^{\prime}_{\upsigma(i)}\vphantom{\big{|}}\right|_{\mathfrak{{p}}^{\prime}_{\upsigma(i)}\times\mathfrak{{p}}^{\prime}_{\upsigma(i)}}}$,
while $d(\upvarphi_{i})_{o_{i}}\colon
T_{o_{i}}M_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}T_{o^{\prime}_{\upsigma(i)}}M^{\prime}_{\upsigma(i)}$
is an isometry with respect to $g_{o_{i}}$ and
$g^{\prime}_{o^{\prime}_{\upsigma(i)}}$. We have a commutative diagram

$\textstyle{\mathfrak{{p}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\scriptstyle{f_{i}}$$\scriptstyle{\sim}$$\textstyle{\mathfrak{{p}}^{\prime}_{\upsigma(i)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{T_{o_{i}}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d(\upvarphi_{i})_{o_{i}}}$$\scriptstyle{\sim}$$\textstyle{T_{o^{\prime}_{\upsigma(i)}}M^{\prime}_{\upsigma(i)}}$

which implies that $\uplambda_{i}=\uplambda^{\prime}_{\upsigma(i)}$. ∎

######  Proposition 21.

Given $1\leqslant i,j\leqslant k$, consider the following conditions:

  1. (i)

$M_{i}$ is isometric to $M_{j}$.

  2. (ii)

$M_{i}$ is homothetic to $M_{j}$.

  3. (iii)

$\mathfrak{{g}}_{i}\simeq\mathfrak{{g}}_{j}$.

Then $(i)\Rightarrow(ii)\Leftrightarrow(iii)$. Moreover, $(ii)\Rightarrow(i)$
if and only if $\uplambda_{i}=\uplambda_{j}$.

###### Proof.

Clearly, $(i)\Rightarrow(ii)$ and, since rescaling the Riemannian metric does
not change the isometry Lie algebra, $(ii)\Rightarrow(iii)$. To show
$(iii)\Rightarrow(ii)$, recall that $M_{i}$ can be recovered from
$\mathfrak{{g}}_{i}$ up to an isometry by taking a simply connected Lie group
$\widehat{G}_{i}$ with $\mathrm{Lie}\hskip
0.59998pt(\widehat{G}_{i})\cong\mathfrak{{g}}_{i}$, a maximal compact subgroup
$\widehat{K}_{i}\subseteq\widehat{G}_{i}$, and endowing the quotient
$\widehat{G}_{i}/\widehat{K}_{i}$ with a suitable $\widehat{G}_{i}$-invariant
Riemannian metric (there is only one up to rescaling). If
$\mathfrak{{g}}_{i}\simeq\mathfrak{{g}}_{j}$, we can always find a Lie group
isomorphism
$F\colon\widehat{G}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\widehat{G}_{j}$
such that $F(\widehat{K}_{i})=\widehat{K}_{j}$, which induces a homothety
$M_{i}\to M_{j}$.

The proof of the last assertion is very similar to that of Lemma 20. Assume
$(ii)$ and start with any homothety $\upvarphi\colon M_{i}\to M_{j}$.
Composing it with a suitable isometry of $M_{j}$ if needed, we can replace
$\upvarphi$ with a homothety $\upvarphi^{\prime}\colon M_{i}\to M_{j}$ mapping
$o_{i}$ to $o_{j}$, and $\upvarphi^{\prime}$ is an isometry if and only if
$\upvarphi$ is. We have an isomorphism
$F\colon\widetilde{{G}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\widetilde{{G}}_{j},\uppsi\mapsto\upvarphi^{\prime}\circ
g\circ\upvarphi^{\prime-1}$, mapping $s_{o_{i}}$ to $s_{o_{j}}$. If we write
$f=F_{*}\colon\mathfrak{{g}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}_{j}$,
then $f(\mathfrak{{p}}_{i})=\mathfrak{{p}}_{j}$, and the following diagram
commutes:

$\textstyle{\mathfrak{{p}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\scriptstyle{f}$$\scriptstyle{\sim}$$\textstyle{\mathfrak{{p}}_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{T_{o_{i}}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\upvarphi^{\prime}_{o_{i}}}$$\scriptstyle{\sim}$$\textstyle{T_{o_{j}}M_{j}}$

The top arrow is an isometry with respect to the inner product
${\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}\times\mathfrak{{p}}_{i}}}$
and
${\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{j}\times\mathfrak{{p}}_{j}}}$.
Consequently, the bottom arrow is an isometry (i.e., $\upvarphi^{\prime}$ is
an isometry) with respect to the inner product
$(g_{i})_{o_{i}}=\uplambda_{i}{\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}\times\mathfrak{{p}}_{i}}}$
on $T_{o_{i}}M_{i}$ and
$(g_{j})_{o_{j}}=\uplambda_{j}{\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{j}\times\mathfrak{{p}}_{j}}}$
on $T_{o_{j}}M_{j}$ if and only if $\uplambda_{i}=\uplambda_{j}$, which
completes the proof. ∎

As we already know, condition $(iii)$ in Proposition 21 is also equivalent to
$\Upsigma_{i}$ and $\Upsigma_{j}$ (or $\mathrm{DD}_{i}$ and $\mathrm{DD}_{j}$)
being weighted-isomorphic. Thus, the symmetric space $M$ is fully determined
up to an isometry by the (weighted) Dynkin diagram $\mathrm{DD}$ of
$\mathfrak{{g}}$ together with the normalizing constants
$\uplambda_{1},\ldots,\uplambda_{k}$ (which we could assign as weights to the
connected components $\mathrm{DD}_{1},\ldots,\mathrm{DD}_{k}$ of
$\mathrm{DD}$). Proposition 21 has the following immediate

######  Corollary 21.1.

The following conditions are equivalent:

  1. (i)

$\uplambda_{i}=\uplambda_{j}$ whenever $M_{i}\sim M_{j}$.

  2. (ii)

$S_{k}^{\simeq}=S_{k}^{\sim}$.

If these conditions are satisfied, we call the Riemannian metric $g$ on $M$
almost Killing. If, moreover, $\uplambda_{1}=\ldots=\uplambda_{k}=1$ (i.e. if
$g_{o}={\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}\times\mathfrak{{p}}}}$),
we call $g$ Killing.

Note that the Riemannian metric of $M$ is automatically almost Killing if $M$
is irreducible. More generally, if no two distinct de Rham factors of $M$ are
homothetic, than its Riemannian metric is almost Killing.

Since the isometry group is not affected by constant rescaling of the metric,
Fact 19 tells us that by rescaling the metric on the de Rham factors of $M$ –
that is, by adjusting the normalizing constants – we might ’gain’ or ’lose’
some connected components of $I(M)$ (whereas $I^{0}(M)$ always stays the
same). From this perspective, the almost Killing condition on the Riemannian
metric ensures precisely that the isometry group is as large as possible,
namely that $I(M)\simeq\left[I(M_{1})\times\cdots\times I(M_{k})\right]\rtimes
S_{k}^{\sim}$.

Having defined the normalizing constants, we can now formulate precisely the
correspondence between symmetric spaces of noncompact type and noncompact real
semisimple Lie algebras that we mentioned at the beginning of the section. Let
$(M,g)$ be a symmetric space of noncompact type with the de Rham decomposition
$M=M_{1}\times\cdots\times M_{k}$. Note that if we rescale the Riemannian
metric on each de Rham factor $M_{i}$ by some constant conformal factor
$a_{i}>0$ and denote the resulting metric on $M$ by $\widetilde{{g}}$, then
$(M,\widetilde{{g}})$ is still a symmetric space of noncompact type and it has
the same isometry Lie algebra. We say that $(M,\widetilde{{g}})$ is obtained
from $(M,g)$ by rescaling the normalizing constants. If $M^{\prime}$ is
another symmetric space of noncompact type, we say that $M$ and $M^{\prime}$
are equivalent if they become isometric after a suitable rescaling of their
normalizing constants. Note that this notion of equivalence is weaker than
being homothetic. We can now formulate the aforementioned correspondence:

$\left\\{\begin{gathered}\text{equivalence classes}\\\ \text{of symmetric
spaces}\\\ \text{of noncompact type}\\\
\end{gathered}\right\\}\xrightarrow[\sim]{M\,\mapsto\,\mathrm{Lie}\hskip
0.59998pt(I(M))}\left\\{\begin{gathered}\text{isomorphism classes}\\\ \text{of
real semisimple Lie algebras}\\\ \text{without nonzero compact ideals}\\\
\end{gathered}\right\\}$

If we start with a real semisimple Lie algebra $\mathfrak{{g}}$ without
nonzero compact ideals, one way to get a symmetric space $M$ of noncompact
type with $\mathrm{Lie}\hskip 0.59998pt(I(M))\simeq\mathfrak{{g}}$ is as
follows. Take a simply connected Lie group $G$ with $\mathrm{Lie}\hskip
0.59998pt(G)\simeq\mathfrak{{g}}$, take a maximal compact subgroup $K\subset
G$, write $M=G/K$, and endow $M$ with any $G$-invariant Riemannian metric. One
can show that $I(M)\cong G/\mathrm{{Z}}(G)$ and thus $\mathrm{Lie}\hskip
0.59998pt(I(M))\simeq\mathfrak{{g}}/\mathfrak{{z}}(\mathfrak{{g}})=\mathfrak{{g}}$.
We already used this construction in the proof of Proposition 21.

We can reformulate part (1) of Proposition 16 in a similar fashion to Fact 19.
For any pair of indices $i,j\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\}$ such that $\mathfrak{{g}}_{i}\simeq\mathfrak{{g}}_{j}$,
pick an isomorphism
$f_{ij}\colon\mathfrak{{g}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{g}}_{j}$
in such a way that if we have
$\mathfrak{{g}}_{i}\simeq\mathfrak{{g}}_{j}\simeq\mathfrak{{g}}_{l}$, then
$f_{jl}\circ f_{ij}=f_{il}$. It follows from Proposition 21 that this gives an
embedding

$\displaystyle S_{k}^{\sim}\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}),\upsigma\mapsto f_{\upsigma},\;\text{where}$
$\displaystyle
f_{\upsigma}(X_{1},\ldots,X_{k})=(f_{\upsigma(1)1}(X_{\upsigma(1)}),\ldots,f_{\upsigma(k)k}(X_{\upsigma(k)}))$

(again, this is an injective group anti-homomorphism). Proposition 16(1) now
asserts that $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{1})\times\cdots\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{k})\right]\rtimes S_{k}^{\sim}$. We have an open
subgroup

$\boldsymbol{\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}}=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{1})\times\cdots\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{k})\right]\rtimes
S_{k}^{\simeq}\subseteq\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}}),$

which may or may not be a proper subgroup, depending on the normalizing
constants777It may be preferable to write $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{(M,g)}$ to avoid ambiguity in case one has another
metric $\widetilde{{g}}$ obtained from $g$ by rescaling the normalizing
constants.. Note that this subgroup does not depend on the choice of
$f_{ij}$’s (although the embedding
$S_{k}^{\sim}\hookrightarrow\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$
surely does depend on this choice). By passing from $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ to $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}$, we are prohibiting those automorphisms of
$\mathfrak{{g}}$ that permute isomorphic simple ideals whose corresponding
normalizing constants do not coincide.

We are now in a position to prove the following result, which characterizes
$I(M)$ intrinsically in terms of the Lie algebra $\mathfrak{{g}}$ – at least
when the metric is almost Killing:

######  Proposition 22.

Let $M$ be a symmetric space of noncompact type. The adjoint map
$\mathrm{Ad}\hskip 0.59998pt\colon I(M)\to\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ is an open embedding of Lie groups with image
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})_{M}$. Moreover,
$\mathrm{Ad}\hskip 0.59998pt$ is an isomorphism if and only if the Riemannian
metric of $M$ is almost Killing. In particular, we always have
$\mathrm{Ad}\hskip 0.59998pt\colon
I^{0}(M)\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{Inn}(\mathfrak{{g}})$.

###### Proof.

To begin with, observe that $\mathrm{Ad}\hskip 0.59998pt$ is a local
isomorphism. Indeed, its induced morphism of Lie algebras is
$\mathrm{ad}\hskip
0.59998pt\colon\mathfrak{{g}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{Der}(\mathfrak{{g}})=\mathrm{Lie}\hskip
0.59998pt(\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}}))$.

Next we prove that $\mathrm{Ad}\hskip 0.59998pt$ is injective. Assume that
$\upvarphi\in\mathrm{Ker}\hskip 0.39993pt(\mathrm{Ad}\hskip 0.59998pt)$. We
first show that $\upvarphi(o)=o$. We have
$(C_{\upvarphi})_{*}=\mathrm{Id}_{\mathfrak{{g}}}$, i.e.
${\left.\kern-1.2ptC_{\upvarphi}\vphantom{\big{|}}\right|_{G}}=\mathrm{Id}_{G}$,
which is the same as to say that $\upvarphi$ commutes with every element of
$G$. In particular, it commutes with every element of $K$, which implies that
$K$ stabilizes $\upvarphi(o)$. Assume that $\upvarphi(o)\neq o$. Then $K$
fixes every point of a geodesic $\upgamma$ emanating from $o$ and passing
through $\upvarphi(o)$. Let $v=\dot{\upgamma}(0)\in T_{o}M$. We see that $v$
is an invariant of the (restricted) isotropy representation of $K$ in
$T_{o}M$. But it is well known that the subspace of invariants of the
(restricted or full) isotropy representation of a symmetric space is contained
in (the tangent space to) the flat factor, which $M$ does not have by
definition. We deduce that $\upvarphi\in K$. But then
$d\upvarphi_{o}={\left.\kern-1.2pt\mathrm{Ad}\hskip
0.59998pt(\upvarphi)\vphantom{\big{|}}\right|_{\mathfrak{{p}}}}=\mathrm{Id}_{\mathfrak{{p}}}$,
so $\upvarphi=e$. This, together with the previous paragraph, implies that
$\mathrm{Ad}\hskip 0.59998pt$ embeds $I(M)$ into $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ as an open subgroup.

Finally, we prove that $\hskip 0.80002pt\mathrm{Im}\hskip
0.39993pt(\mathrm{Ad}\hskip 0.59998pt)=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}$ (note that the rest will follow, as this
subgroup equals the whole $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ if
and only if $S_{k}^{\simeq}=S_{k}^{\sim}$, i.e. if and only if the metric is
almost Killing). Recall that if $V$ is a (finite-dimensional) vector space,
then the representation of $\mathrm{GL}(V)$ in $V$ admits a unique extension
to a representation in the full tensor algebra
$TV=\bigoplus_{p,q=0}^{\infty}T^{(p,q)}V$ by algebra automorphisms such that
on $V^{*}$ it coincides with the dual representation. We will make use of the
following description of the isotropy representation (see exercise A.6 on p.
227 in [6] for the statement and p. 564 there for a solution):

######  Fact 23.

Let $M$ be a simply connected symmetric space, $o\in M$, and
$\widetilde{{K}}\subseteq I(M)$ the full isotropy subgroup at $o$. Then
$T\in\mathrm{GL}(T_{o}M)$ lies in the image of the isotropy representation
$\widetilde{{K}}\hookrightarrow\mathrm{GL}(T_{o}M)$ if and only if it
preserves the inner product888This first condition cuts out precisely
$\mathrm{O}(T_{o}M)$. $g_{o}\in T^{(0,2)}T_{o}M$ and the curvature tensor
$R_{o}\in T^{(1,3)}T_{o}M$.

Fix $o\in M$ as before and consider the Cartan involution
$\uptheta=\mathrm{Ad}\hskip 0.59998pt(s_{o})$ and the subgroup
$\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}\subseteq\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ (we used to denote it by $K$ in Section 3). We want
to show that $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{\uptheta}$
intersects every connected component of $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$. Take any $\upeta\in\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$. The automorphism $\upeta\uptheta\upeta^{-1}$ is
also a Cartan involution. Since all Cartan involutions are conjugate by inner
automorphisms, there exists
$\updelta\in\mathrm{Inn}(\mathfrak{{g}})=\mathrm{Aut}\hskip
0.59998pt^{0}(\mathfrak{{g}})$ such that
$\updelta\upeta\uptheta\upeta^{-1}\updelta^{-1}=\uptheta$, i.e.
$\updelta\upeta\in\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})^{\uptheta}$. As
$\updelta\upeta$ and $\upeta$ lie in the same connected component of
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$, we are done.

It then suffices to show that $\hskip 0.80002pt\mathrm{Im}\hskip
0.39993pt(\mathrm{Ad}\hskip 0.59998pt)$ contains999In order for the RHS in
this expression to make sense, we need to put an additional requirement on the
choice of $f_{ij}$’s, namely $f_{ij}\circ\uptheta_{i}=\uptheta_{j}\circ
f_{ij}$ for any $i,j$.

$\boldsymbol{\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}^{\uptheta}}=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}\cap\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}=\left[\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{1})^{\uptheta_{1}}\times\cdots\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{k})^{\uptheta_{k}}\right]\rtimes S_{k}^{\simeq}.$

Take any element $\upeta$ of this subgroup. It preserves the Cartan
decomposition $\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$, so we can
write
$T={\left.\kern-1.2pt\upeta\vphantom{\big{|}}\right|_{\mathfrak{{p}}}}\in\mathrm{GL}(\mathfrak{{p}})\cong\mathrm{GL}(T_{o}M)$.
We claim that $T$ lies in the image of the isotropy representation
$\widetilde{{K}}\hookrightarrow\mathrm{O}(T_{o}M)$. Due to Fact 23, it
suffices to show that it is orthogonal and preserves the curvature tensor. As
any other automorphism of $\mathfrak{{g}}$, $\upeta$ is orthogonal with
respect to $B$, so $T$ is orthogonal with respect to
${\left.\kern-1.2ptB\vphantom{\big{|}}\right|_{\mathfrak{{p}}\times\mathfrak{{p}}}}$.
By construction, for every $i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\}$, if we write $T(\mathfrak{{p}}_{i})=\mathfrak{{p}}_{j}$,
then $\uplambda_{i}=\uplambda_{j}$, so
${\left.\kern-1.2ptT\vphantom{\big{|}}\right|_{\mathfrak{{p}}_{i}}}\colon\mathfrak{{p}}_{i}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{p}}_{j}$
is an isometry with respect to the inner products $(g_{i})_{o_{i}}$ and
$(g_{j})_{o_{j}}$, which implies that $T$ is orthogonal with respect to
$g_{o}$ as well.

The fact that $T$ preserves the curvature tensor at $o$ can be readily seen
from the well-known expression for $R_{o}$ under the identification
$T_{o}M\cong\mathfrak{{p}}$: if $X,Y,Z\in\mathfrak{{p}}$ then
$R_{o}(X,Y)Z=-[[X,Y],Z]$. We deduce that there exists $k\in\widetilde{{K}}$
such that $\mathrm{Ad}\hskip 0.59998pt(k)$ and $\upeta$ coincide on
$\mathfrak{{p}}$. Since they are both Lie algebra automorphisms, they have to
coincide on the subspace
$[\mathfrak{{p}},\mathfrak{{p}}]\subseteq\mathfrak{{k}}$ as well. The rest
follows from the following

######  Lemma 24.

Let $\mathfrak{{g}}$ be a real semisimple Lie algebra without nonzero compact
ideals, and let $\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$ be a
Cartan decomposition. Then $[\mathfrak{{p}},\mathfrak{{p}}]=\mathfrak{{k}}$.

###### Proof of the lemma.

If $\mathfrak{{g}}$ is simple, this is the content of [7, Ch. VI, problems
22-24] (see p. 735 for solutions). If $\mathfrak{{g}}$ is the sum of several
noncompact simple ideals, namely
$\mathfrak{{g}}=\bigoplus_{i=1}^{k}\mathfrak{{g}}_{i}$, then
$\mathfrak{{k}}=\bigoplus_{i=1}^{k}\mathfrak{{k}}_{i},\mathfrak{{p}}=\bigoplus_{i=1}^{k}\mathfrak{{p}}_{i}$,
where
$\mathfrak{{k}}_{i}=\mathfrak{{k}}\cap\mathfrak{{g}}_{i},\mathfrak{{p}}_{i}=\mathfrak{{p}}\cap\mathfrak{{g}}_{i}$,
so we have $[\mathfrak{{p}}_{i},\mathfrak{{p}}_{i}]=\mathfrak{{k}}_{i}$ and
the assertion follows. ∎

∎

Looking at the proof of Proposition 22, we have the following

######  Corollary 22.1.

In the notation of Proposition 22, the map $\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}\colon\widetilde{{K}}\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$ is an open embedding of Lie groups with
image $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})_{M}^{\uptheta}$. Moreover,
its image is the whole $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$ if and only if the Riemannian metric is
almost Killing. In particular, we always have $\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}\colon
K\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{Inn}(\mathfrak{{g}})^{\uptheta}$
and $\widetilde{{K}}=\widetilde{{G}}^{\Uptheta}$ (and thus $K=G^{\Uptheta}$).

###### Proof.

We need only prove the very last assertion. We compute:

$\displaystyle\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}(\widetilde{{G}}^{\Uptheta})$
$\displaystyle=\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}(\widetilde{{G}})^{\uptheta}$
$\displaystyle(\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}\circ\Uptheta=C_{\uptheta}\circ\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}})$ $\displaystyle=\mathrm{Ad}\hskip
0.59998pt(\widetilde{{G}})\cap\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$ $\displaystyle=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}\cap\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$ $\displaystyle(\text{by Proposition
\ref{isometry_automorphism}})$ $\displaystyle=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})_{M}^{\uptheta}$ $\displaystyle=\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}(\widetilde{{K}})$ $\displaystyle(\text{by the
first assertion}),$

so $\widetilde{{G}}^{\Uptheta}=\widetilde{{K}}$. ∎

We can use Proposition 22 to prove the following simple result, which tells
how one can recover $M$ from its isometry Lie algebra $\mathfrak{{g}}$ in an
invariant fashion. Given $\mathfrak{{g}}$ real semisimple, note that the set
$\mathcal{C}\subseteq\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ of all
Cartan involutions on $\mathfrak{{g}}$ is an immersed submanifold of
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$, since it is an orbit of the
adjoint action of $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ on itself.

######  Proposition 25.

Let $M$ be a symmetric space of noncompact type and
$\mathfrak{{g}}=\mathrm{Lie}\hskip 0.59998pt(I(M))$. Then the map $\Upxi\colon
M\to\mathcal{C},p\mapsto\mathrm{Ad}\hskip 0.59998pt(s_{p})$, is a
diffeomorphism.

Note that we do not assume the Riemannian metric to be almost Killing here.

###### Proof.

We know that each $\mathrm{Ad}\hskip 0.59998pt(s_{p})$ is indeed a Cartan
involution. If we identify $I(M)$ with a subgroup of $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ by means of $\mathrm{Ad}\hskip 0.59998pt$, $\Upxi$
is easily seen to be $I(M)$-equivariant. As both $M$ and $\mathcal{C}$ are
smooth homogeneous $I(M)$-spaces, $\Upxi$ is smooth and surjective. Since we
already know that $\mathrm{Ad}\hskip 0.59998pt$ is injective, it suffices to
show that the map $M\to I(M),p\mapsto s_{p}$, is injective. Given $p\in M$,
observe that $p$ is the only fixed point of the symmetry $s_{p}$ (this is
specific to symmetric spaces of noncompact type and utterly fails for
symmetric spaces of compact type). Therefore, the symmetries $s_{p}$ and
$s_{q}$ cannot coincide unless $p=q$, which completes the proof. ∎

Combining Propositions 16(1) and 22, we obtain the following:

######  Corollary 26.

Let $\mathfrak{{g}}$ be a real semisimple Lie algebra. For any Cartan
involution $\uptheta$ on $\mathfrak{{g}}$, $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}$ is a maximal compact subgroup of
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})$ and
$\mathrm{Inn}(\mathfrak{{g}})^{\uptheta}$ is a maximal compact subgroup of
$\mathrm{Inn}(\mathfrak{{g}})$.

###### Proof.

Let us write
$\mathfrak{{g}}=\mathfrak{{g}}_{\mathrm{{c}}}\oplus\mathfrak{{g}}_{\mathrm{{nc}}}$,
where $\mathfrak{{g}}_{\mathrm{{c}}}$ is the sum of all compact ideals of
$\mathfrak{{g}}$ and $\mathfrak{{g}}_{\mathrm{{nc}}}$ is the complementary
noncompact semisimple ideal. We know from Section 3 that $\uptheta$ respects
this decomposition and is the identity on $\mathfrak{{g}}_{\mathrm{{c}}}$, so
we can write
$\uptheta=\mathrm{Id}_{\mathfrak{{g}}_{\mathrm{{c}}}}+\uptheta_{\mathrm{{nc}}}$.
According to Proposition 16(1), $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})$, so $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta}=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})\times\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})^{\uptheta_{\mathrm{{nc}}}}$. Take a
symmetric space $M$ of noncompact type with $\mathrm{Lie}\hskip
0.59998pt(I(M))\cong\mathfrak{{g}}_{\mathrm{{nc}}}$ and $o\in M$ such that
$\mathrm{Ad}\hskip 0.59998pt(s_{o})=\uptheta_{\mathrm{{nc}}}$ (the latter is
possible by Proposition 25). Rescale the normalizing constants of $M$ if
necessary (this would change neither the isometry Lie algebra nor $s_{o}$) so
that the metric becomes almost Killing. By Proposition 22 and Corollary 22.1,
we have $\mathrm{Ad}\hskip 0.59998pt\colon
I(M)\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})$ and $\mathrm{Ad}\hskip
0.59998pt_{\widetilde{{G}}}(\widetilde{{K}})=\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})^{\uptheta_{\mathrm{{nc}}}}$, which
implies that $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})^{\uptheta_{\mathrm{{nc}}}}$ is a
maximal compact subgroup of $\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}}_{\mathrm{{nc}}})$. We are left to prove that
$\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}}_{\mathrm{{c}}})$ is compact. This
simply follows from the fact that it is a closed subgroup of
$O_{B_{\mathrm{{c}}}}(\mathfrak{{g}}_{\mathrm{{c}}})$, and the latter is
compact because the Killing form $B_{\mathrm{{c}}}$ of
$\mathfrak{{g}}_{\mathrm{{c}}}$ is negative-definite. The last assertion
follows easily, for $(\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})^{\uptheta})^{0}=\mathrm{Aut}\hskip
0.59998pt^{0}(\mathfrak{{g}})^{\uptheta}=\mathrm{Inn}(\mathfrak{{g}})^{\uptheta}$
has to be a maximal compact subgroup of $\mathrm{Aut}\hskip
0.59998pt^{0}(\mathfrak{{g}})=\mathrm{Inn}(\mathfrak{{g}})$. ∎

Eventually, we can reformulate the results of Section 3 in the language of
symmetric spaces. Let $M$ be a symmetric space of noncompact type, and let us
keep all the notation as above. Just as we did in Section 3, take
$\mathfrak{{a}}\subseteq\mathfrak{{p}}$ a maximal abelian subspace,
$\Upsigma\subseteq\mathfrak{{a}}^{*}$ the corresponding restricted root
system, and $\mathrm{DD}$ its Dynkin diagram (for some choice $\Upsigma^{+}$).
Let $M=M_{1}\times\cdots\times M_{k}$ be the de Rham decomposition of $M$. In
a similar fashion to $\mathrm{Aut}\hskip 0.59998pt(\mathfrak{{g}})_{M}$, we
can define:

$\displaystyle\boldsymbol{\mathrm{Aut}\hskip
0.59998pt^{\mathrm{{w}}}(\Upsigma)_{M}}$
$\displaystyle\vcentcolon=\prod_{i=1}^{k}\mathrm{Aut}\hskip
0.59998pt(\Upsigma_{i})\rtimes
S_{k}^{\simeq}\,\subseteq\,\prod_{i=1}^{k}\mathrm{Aut}\hskip
0.59998pt(\Upsigma_{i})\rtimes S_{k}^{\sim}\simeq\mathrm{Aut}\hskip