# On extensions of the Jacobson–Morozov theorem to even characteristic

David I. Stewart  Department of Mathematics, Alan Turing Building, Manchester,
M13 9PL, UK<EMAIL_ADDRESS>and  Adam R. Thomas  Department
of Mathematics, University of Warwick, Coventry, CV4 7AL, UK
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a simple algebraic group over an algebraically closed field $\Bbbk$
of characteristic $2$. We consider analogues of the Jacobson–Morozov theorem
in this setting. More precisely, we classify those nilpotent elements with a
simple $3$-dimensional Lie overalgebra in
$\mathfrak{g}:=\operatorname{Lie}(G)$ and also those with overalgebras
isomorphic to the algebras $\operatorname{Lie}(\mathrm{SL}_{2})$ and
$\operatorname{Lie}(\mathrm{PGL}_{2})$. This leads us to calculate the
dimension of Lie automiser $\mathfrak{n}_{\mathfrak{g}}(\Bbbk\cdot
e)/\mathfrak{c}_{\mathfrak{g}}(e)$ for all nilpotent orbits; in even
characteristic this quantity is very sensitive to isogeny.

###### 2010 Mathematics Subject Classification:

17B45

##  1\. Introduction

The Jacobson–Morozov theorem states that every nilpotent element $e$ in the
Lie algebra $\mathfrak{g}$ of a semisimple complex algebraic group $G$ can be
extended to an $\mathfrak{sl}_{2}$-triple $(e,h,f)\in\mathfrak{g}^{3}$
satisfying the famous commutator relations

$[h,e]=2e,\ [h,f]=-2f,\ \text{and }[e,f]=h.$

In recent work [ST18] building on previous efforts by Pommerening, Premet and
Kostant—among others—the authors gave an exhaustive account of when the
statement of the theorem holds for a reductive algebraic group $G$ over an
algebraically closed field $\Bbbk$ of characteristic $p\geq 3$, with sharp
statements about uniqueness of the embedding up to conjugacy. It turned out
that _all_ nilpotent elements can be embedded into $\mathfrak{sl}_{2}$-triples
with essentially one exception: in $G_{2}$ when $p=3$, a nilpotent element in
the exceptional orbit (usually labelled $\tilde{A}_{1}^{(3)}$) lives in no
subalgebra isomorphic to $\mathfrak{sl}_{2}$.

As can be seen from the commutator relations above, there is some awkwardness
in finding the right analogue of the statement of the Jacobson–Morozov theorem
in characteristic $2$ and there seem to be three reasonable approaches.
Firstly, one may take the straight modulo $2$ reduction of the relations
($\star$) to get the Lie algebra
$\mathfrak{sl}_{2}=\operatorname{Lie}(\mathrm{SL}_{2})$, in which $h$ is
central, so that $\mathfrak{sl}_{2}$ is no longer simple, and indeed the
reduced elements $(\bar{e},\bar{h},\bar{f})$ generate a (nilpotent)
Heiseinberg algebra. Taking instead a minimal lattice with $[h,e]=e$,
$[h,f]=f$ and $[e,f]=2h$, then the reduction modulo $2$ gets generators and
relations for the ‘dual’ Lie algebra
$\mathfrak{pgl}_{2}=\operatorname{Lie}(\mathrm{PGL}_{2})$, which is evidently
still not simple: it has a $2$-dimensional abelian $p$-nilpotent ideal
$\langle e,f\rangle$ on which $\operatorname{ad}(h)$ acts as the identity.
However, setting $[h,e]=e$, $[h,f]=f$ and $[e,f]=h$ we do indeed get a simple
Lie algebra $\mathfrak{s}$ of dimension $3$, which is unique up to
isomorphism.111Some prefer the cyclic basis $[x,y]=z$, $[y,z]=x$, $[z,x]=y$.
Colloquially, the Lie algebra $\mathfrak{s}$ is sometimes referred to as _fake
$\mathfrak{sl}_{2}$_.

Let us give a little more information about $\mathfrak{s}$. On the one hand,
we may identify $\mathfrak{s}$ with the Cartan type Zassenhaus Lie algebra
$W(1;2)^{\prime}=\langle\partial,x^{(1)}\partial,x^{(2)}\partial\rangle$ in
characteristic $2$; see [SF88] for a definition, using divided powers. On the
other hand, one may see $\mathfrak{s}$ as part of a larger family of simple
orthogonal Lie algebras in characteristic $2$ analogous to those in type $B$.
Following [BGL09, 4.6.1] one sets $\hat{\mathfrak{o}}_{2n+1}$ to be the Lie
algebra defined by a Cartan matrix of type $B_{n}$, after modifying the
commutator relations by halving the end column222We use the Bourbaki notation
[Bou05], so the long roots are on the left of the Dynkin diagram.. The
resulting commutator relations define a simple Lie algebra; indeed we have
$\mathfrak{s}\cong\hat{\mathfrak{o}}_{3}$. It is a simple calculation to see
that the algebra $\mathfrak{s}$ is not restricted: it has codimension $2$ in
its minimal $p$-envelope, $\mathfrak{s}_{p}$, with basis $\\{E,e,h,f,F\\}$
where $[E,F]=h$, $[E,f]=e$, $[F,e]=f$ and $h$ centralises $E$ and $F$. We note
that the irreducible representations of $\mathfrak{s}$ were described in
[Dol78].

The statements of our results are given in terms of the description of
nilpotent elements in the Lie algebras of classical groups in terms of certain
standard forms, developed by Hesselink [Hes79] and presented in [LS12].
Moreover, our proofs rely on explicit representatives of these classes as sums
of root vectors, utilising some earlier work [KST22]. Since extracting the
representative of a general class from _op. cit._ is not entirely
straightforward, we suspect readers will appreciate the very explicit formulas
for the representatives which we give in Section 2.3 below.

Theorems 4.1, 5.1 and 6.1 give a complete list of those nilpotent orbits
$G(k)\cdot e$ such that $e$ lies in a subgroup isomorphic with $\mathfrak{s}$,
$\mathfrak{pgl}_{2}$ and $\mathfrak{sl}_{2}$ respectively. When
$e\in\mathfrak{pgl}_{2}$ or $e\in\mathfrak{s}$, there is an element $h$ such
that $[h,e]=e$. And when such an $h$ exists, the Lie-theoretic normaliser
$\mathfrak{n}_{\mathfrak{g}}(\Bbbk\cdot e)$ is stricly larger than the
centraliser $\mathfrak{c}_{\mathfrak{g}}(\Bbbk\cdot e)$; another way to say
this is that the Lie-theoretic automiser
$\mathfrak{n}_{\mathfrak{g}}(\Bbbk\cdot e)/\mathfrak{c}_{\mathfrak{g}}(e)$ is
non-trivial (and must have dimension $1$). In Theorem 3.1, we give a complete
list of the orbits where this happens, whose members turn out to be
surprisingly sporadic. We then use that result to inform the proofs of
Theorems 4.1 and 5.1.

When $p>2$, the automiser of $\Bbbk\cdot e$ is always $1$-dimensional: either
$e$ lies in an $\mathfrak{sl}_{2}$-subalgebra; or $p=3$, $G\cong G_{2}$ and
$e$ has label $A_{1}^{(3)}$ where one can do a direct check. Hence the
calculation of the automiser is only interesting when $p=2$, and Theorem 3.1
then answers that line of enquiry.

For other recent work in this direction, see [GP, PT23].

### Acknowledgments

The first author is supported by a Leverhulme Trust Research Project Grant
RPG-2021-080 and the second author is supported by an EPSRC grant
EP/W000466/1. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

##  2\. Preliminaries and notation

###  2.1. A presentation for $\mathfrak{g}$

Throughout, $\Bbbk$ will be an algebraically closed field of characteristic
$2$ and $G$ will be a simple algebraic $\Bbbk$-group. Let $T$ be a maximal
torus of $G$ of rank $n$, giving rise to a root datum
$(X(T),\Phi,Y(T),\Phi^{\vee})$ that determines $G$ up to isomorphism. Recall
this means $X(T)=\mathrm{Hom}(T,\mathbb{G}_{m})$ is the weight lattice, which
is a free $\mathbb{Z}$-module of rank $n$ and $Y(T)$ is the
$\mathbb{Z}$-linear dual space $\mathrm{Hom}(\mathbb{G}_{m},T)$, which come
with a perfect pairing $\langle\\_,\\_\rangle:X(T)\times Y(T)\to\mathbb{Z}$;
the set of roots $\Phi\subseteq X(T)$ is the set of non-zero weights of $T$ on
$\mathfrak{g}=\operatorname{Lie}(G)$; and $\alpha\mapsto\alpha^{\vee}$ is a
map from $\Phi$ into $Y(T)$ with image denoted $\Phi^{\vee}$, such that
$s_{\alpha}(\beta)=\beta-\langle\beta,\alpha^{\vee}\rangle$—in particular,
$\langle\alpha,\alpha^{\vee}\rangle=2$ and
$\langle\alpha,\beta^{\vee}\rangle\in\mathbb{Z}$.

The Cartan matrix is
$\mathsf{C}=(\langle\alpha_{i},\alpha_{j}^{\vee}\rangle)_{ij}$. A choice of
basis $\\{\omega_{1},\dots,\omega_{n}\\}$ for the lattice $X(T)$ is equivalent
to a factorisation

(1) $C=\mathsf{A}\mathsf{B}^{T},$

where $\mathsf{A}=(\langle\alpha_{i},\omega_{j}^{\vee}\rangle)_{ij}$ and
$\mathsf{B}=(\langle\omega_{i},\alpha_{j}^{\vee}\rangle)_{ij}$. In Table 1 we
give choices for $\mathsf{A}$ and $\mathsf{B}$ in certain cases.

Description of $G$ | $\mathsf{A}$ | $\mathsf{B}$
---|---|---
simply connected | $\mathsf{C}$ | $I_{n}$
adjoint | $I_{n}$ | $\mathsf{C}$
$\mathrm{SO}_{2n}$ | $\begin{bmatrix}1&0&0&\dots&0&0&0\\\ 0&1&0&\dots&0&0&0\\\ 0&0&1&\dots&0&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&0&0&\dots&1&0&0\\\ 0&0&0&\dots&0&1&0\\\ 0&0&0&\dots&0&1&2\end{bmatrix}$ | $\begin{bmatrix}2&-1&0&\dots&0&0&0\\\ -1&2&-1&\dots&0&0&0\\\ 0&-1&2&\dots&0&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&0&0&\dots&2&-1&0\\\ 0&0&0&\dots&-1&2&-1\\\ 0&0&0&\dots&-1&0&1\end{bmatrix}$
$\mathrm{HSpin}_{2n}$ ($n$ even) | $\begin{bmatrix}1&0&0&0&\dots&0&0&0&0\\\ 0&1&0&0&\dots&0&0&0&0\\\ 0&0&1&0&\dots&0&0&0&0\\\ 0&0&0&1&\dots&0&0&0&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&0&0&\dots&1&0&0&0\\\ 0&0&0&0&\dots&0&1&0&0\\\ 1&0&1&0&\dots&1&0&2&0\\\ 0&0&0&0&\dots&0&0&0&1\end{bmatrix}$ | $\begin{bmatrix}2&-1&0&0&\dots&0&0&-1&0\\\ -1&2&-1&0&\dots&0&0&1&0\\\ 0&-1&2&-1&\dots&0&0&-1&0\\\ 0&0&-1&2&\dots&0&0&1&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&0&0&\dots&2&-1&-1&0\\\ 0&0&0&0&\dots&-1&2&0&-1\\\ 0&0&0&0&\dots&0&-1&1&0\\\ 0&0&0&0&\dots&0&-1&0&2\end{bmatrix}$
Table 1. Choices of factorisations $\mathsf{C}=\mathsf{A}\mathsf{B}^{T}$ of
the Cartan matrix. We take $\mathsf{A}$ as in [Roo10, 4.4.4] except in for
$\mathrm{HSpin}_{2n}$, where we swapped the roles of $\alpha_{n-1}$ and
$\alpha_{n}$ for later convenience.

Some explicit calculation with root vectors in $\mathfrak{g}$ are unavoidable
and so we fix Chevalley bases following [CR09]. This means giving a
presentation of an integral form $\mathfrak{g}_{\mathbb{Z}}$ of $\mathfrak{g}$
as follows. Let
$\mathfrak{h}_{\mathbb{Z}}:=\langle\mathsf{h}_{1},\dots,\mathsf{h}_{n}\rangle$
be a free $\mathbb{Z}$-module of rank $n$ and take the direct sum of
$\mathfrak{h}_{\mathbb{Z}}$ with a free $\mathbb{Z}$-module of rank $|\Phi|$
to form

$\mathfrak{g}_{\mathbb{Z}}:=\mathbb{Z}\mathsf{h}_{1}\oplus\dots\oplus\mathbb{Z}\mathsf{h}_{n}\oplus\bigoplus_{\alpha\in\Phi}\mathbb{Z}\mathsf{e}_{\alpha}.$

We then make $\mathfrak{g}_{\mathbb{Z}}$ into a Lie algebra by giving its
structure constants. For all $1\leq i,j\leq n$ and $\alpha,\beta\in\Phi$, let:

(CB1) $\displaystyle[\mathsf{h}_{i},\mathsf{h}_{j}]=0;$ (CB2)
$\displaystyle[\mathsf{h}_{i},\mathsf{e}_{\alpha}]=\langle\alpha,\omega_{i}^{\vee}\rangle;$
(CB3)
$\displaystyle[\mathsf{e}_{\alpha},\mathsf{e}_{-\alpha}]=\sum_{i=1}^{n}\langle\omega_{i},\alpha^{\vee}\rangle\mathsf{h}_{i};$
(CB4)
$\displaystyle[\mathsf{e}_{\alpha},\mathsf{e}_{\beta}]:=\begin{cases}N_{\alpha,\beta}\mathsf{e}_{\alpha+\beta}&\text{
if }\alpha+\beta\in\Phi,\\\ 0&\text{otherwise.}\end{cases}$

The integer $N_{\alpha+\beta}$ is defined to be
$\epsilon_{\alpha\beta}p_{\alpha,\beta}+1$ where $p_{\alpha,\beta}$ is the
largest integer such that $\beta-p_{\beta,\gamma}\gamma\in\Phi$. (The
coefficient $\epsilon_{\alpha\beta}$ is always $\pm 1$, so in $\Bbbk$ it may
be ignored.)

It is a theorem of Chevalley [CR09, Thm. 2] that (CB1)–(CB4) determine a
presentation of $\mathfrak{g}$. More precisely, there is an isomorphism of Lie
algebras $\mathfrak{g}_{Z}\otimes_{\mathbb{Z}}\Bbbk\cong\mathfrak{g}$; and
under that isomorphism the abelian subalgebra
$\mathfrak{h}_{\mathbb{Z}}\otimes\Bbbk$ is sent to the Cartan subalgebra
$\mathfrak{h}$, and $\mathbb{Z}\mathsf{e}_{\alpha}\otimes_{\mathbb{Z}}\Bbbk$
goes to $\Bbbk\cdot\mathsf{e}_{\alpha}$.

The equations above imply that $\mathfrak{g}$ has a grading by root height.
Given a root $\beta=\sum_{i=1}^{n}a_{i}\alpha_{i}\in\Phi$, the height of
$\gamma$ is $\text{ht}(\gamma)=\sum_{i=1}^{n}a_{i}$. The roots of positive
height are the positive roots $\Phi^{+}$ and those of negative height are the
negative roots $\Phi^{-}$. Define also $\tilde{\alpha}$ to be the highest root
in $\Phi$. We have:

$\displaystyle\mathfrak{g}$
$\displaystyle=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}_{i}\quad\text{ is a
grading of $\mathfrak{g}$, where
}\quad\mathfrak{g}_{0}=\mathfrak{h}\quad\text{ and
}\quad\mathfrak{g}_{i}=\bigoplus_{\mathrm{ht}(\alpha)=i}\Bbbk e_{\alpha}$

We also write $\mathfrak{g}_{\geq i}:=\bigoplus_{\text{ht}(\alpha)\geq
i}\mathfrak{g}_{\alpha}$ with $\mathfrak{g}_{\leq i}$ defined similarly. Put
$\mathfrak{g}_{+}:=\mathfrak{g}_{\geq 0}$,
$\mathfrak{g}_{-}:=\mathfrak{g}_{\leq 0}$. And lastly set $\pi_{i}(x)$ be the
projection of $x$ to $\mathfrak{g}_{i}$.

We record some formulas that follow from the choices for $\mathsf{A}$ in Table
1. For brevity we write

$e_{i}:=e_{\alpha_{i}}\qquad\text{ and }\qquad
f_{i}:=e_{-\alpha_{i}}\qquad\text{ for }\alpha_{i}\in\Delta.$

When $G$ is adjoint, we have $\mathsf{A}=I$ and so if
$h:=\sum_{i=1}^{n}h_{i}$, we get

(2) $[h,e_{\gamma}]=\mathrm{ht}(\gamma).$

When $G=\mathrm{SO}_{2n}$, we have $1\leq i\leq n$ and all roots
$\gamma=\sum_{j}c_{j}\alpha_{j}$

(3)
$\displaystyle[h_{i},e_{\gamma}]=\left(\left(\sum_{j=1}^{n-2}\delta_{i,j}c_{j}\right)+\delta_{i,n-1}(c_{n-1}+c_{n})\right)e_{\gamma}.$

When $G=\mathrm{HSpin}_{2n}$, for $1\leq i\leq n-2$ set
$\epsilon(i)=i\pmod{2}$ and set $\epsilon(n-1)=\epsilon(n)=0$. Then for $1\leq
i\leq n$ and all roots $\gamma=\sum_{j}c_{j}\alpha_{j}$ we have

(4)
$\displaystyle[h_{i},e_{\gamma}]=\left(\left(\sum_{j=1}^{n-2}\delta_{i,j}c_{j}\right)+\delta_{i,n-1}c_{n-1}+\epsilon(i)c_{n-1}\right)e_{\gamma}.$

######  Remark 2.1.

Our choice of matrix $\mathsf{A}$ when $G\cong\mathrm{HSpin}_{2n}$ slightly
obscures the useful fact that if one restricts to the Levi subsystem
corresponding to the nodes $1,\ldots,\alpha_{n-2},\alpha_{n}$, then the
corresponding $n-1\times n-1$ minor of $\mathsf{A}$ is an identity matrix.

###  2.2. Nilpotent orbits

Since char $\Bbbk>0$, the Lie algebra $\mathfrak{g}$ comes with a $[p]$ map,
$x\mapsto x^{[p]}$, which is $p$-semilinear in $\Bbbk$ and satisfies
$\operatorname{ad}(x^{[p]})=(\operatorname{ad}(x))^{p}$. An element
$x\in\mathfrak{g}$ is _nilpotent_ if $x^{[p]^{r}}=0$ for some $r\geq 0$; it is
_semisimple_ if $x$ is in the $\Bbbk$-span of $x^{[p]},\dots,x^{[p]^{r}}$ for
some $r\geq 0$; it is _toral_ if $x^{[p]}=x$; and a subalgebra of
$\mathfrak{g}$ with a basis of toral elements is called a _torus_. Since
$\mathfrak{g}=\operatorname{Lie}(G)$ for $G$ simple, its Cartan subalgebras
are maximal tori. The properties above are invariant under conjugacy by $g\in
G(k)$. In particular, an orbit $\mathcal{O}_{e}=G(k)\cdot e$ of elements of
$\mathfrak{g}$ is called a nilpotent orbit if it consists of nilpotent
elements. The union of all nilpotent orbits of $\mathfrak{g}$ is denoted
$\mathcal{N}:=\mathcal{N}(\mathfrak{g})$ and called the _nilpotent cone_.

For the general theory of nilpotent orbits and algebro-geometric properties of
$\mathcal{N}$ we refer the reader to [Jan04]. For complete data of the
classification of the nilpotent orbits we recommend [LS04], but we give a
digest of the salient points below.

###  2.3. Standard forms and representatives for nilpotent classes for
classical groups

Let $G$ be classical. Suppose that $\phi:G\rightarrow H$ is a central isogeny.
Then $d\phi$ induces a bijection between the nilpotent cones of
$\operatorname{Lie}(G)$ and $\operatorname{Lie}(H)$; for details, see [Jan04,
Prop. 2.7]. Therefore when enumerating nilpotent orbits it is sufficient for
us to assume $G$ is one of $\mathrm{SL}(V)$, $\mathrm{Sp}(V)$ or
$\mathrm{SO}(V)$ and we refer to $V$ as the natural module for $G$.

A _standard form_ for a nilpotent element $e$ of
$\mathfrak{g}=\operatorname{Lie}(G)$ is given in [LS12]. It consists of
listing indecomposable summands occurring in the restriction of $V$ to
$\Bbbk[e]$ predicated on the bilinear or quadratic form that $G$ preserves on
$V$; ultimately the possibilities are parametrised by a small number of
decreasing sequences of positive integers $\mathcal{S}(e)$ that we call the
_specification_ of $e$. It is a theorem that the orbit of $e$ is determined by
$\mathcal{S}(e)$, with one collection of exceptions—see Remark 2.3. The
possible standard forms and their specifications are described in Table 2.

In [KST22], the authors give a recipe for writing down a representative of $e$
in terms of root vectors given the specification $\mathcal{S}(e)$. Table
LABEL:tab:stdreps works out this recipe explicitly in all cases. The main tool
for generating representatives is the Borel–de Siebenthal algorithm, which is
a well-known procedure for finding bases for root subsystems; here a _root
subsystem_ $\Psi$ is a subset of $\Phi$ which is symmetric in the sense that
$\Psi=-\Psi$ and so that $g_{\Psi}:=\langle
e_{\psi},h_{i}\mid\psi\in\Psi,1\leq i\leq n\rangle$ is a Lie subalgebra with
root system $\Psi$. In Table LABEL:tab:stdreps we give for each $e$ a natural
subsystem $\Psi:=\Psi(e)$ such that $e\in g_{\Psi}$. The subsystem $\Psi(e)$
has a base $\Delta_{\Psi}$ which is very nearly the set of roots $J:=J_{1}\cup
J_{2}$ on which $e$ is supported—the table explains how to modify $J_{2}$ to
$J_{2}^{\prime}$ so that $\Delta_{\Psi}=J_{1}\cup J_{2}^{\prime}$.

For the interested reader, we explain the modification of $J_{2}$ to
$J_{2}^{\prime}$ in a representative example where $\Phi$ has type $B_{n}$. If
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},m)$,
then

$\Psi=A_{k_{1}-1}\dots A_{k_{r}-1}D_{m_{1}}\dots D_{m_{s}}B_{m-1}.$

Applying the Borel–de Siebenthal algorithm gives a base for the $D_{m_{i}}$
factor as

$\alpha_{v_{i}+m_{i}-1},\ldots,\alpha_{v_{i}+1},-\tilde{\alpha}_{i}=-(\alpha_{v_{i}+1}+2\alpha_{v_{i}+1}+\cdots+2\alpha_{n}),$

where $\tilde{\alpha}_{i}$ is the highest root of the type $B$ subsystem with
base $\\{\alpha_{v_{i}+1},\dots,\alpha_{n}\\}$, and the integer $v_{i}$ is
defined in Table LABEL:tab:stdreps. A straightforward Weyl group calculation
shows that

$\alpha_{v_{i}+1},\ldots,\alpha_{v_{i}+m_{i}-1},\ \
\widetilde{\beta_{i}}=\alpha_{v_{i+1}-1}+2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-1}+2\alpha_{n}$

is also a base for the given $D_{m_{i}}$ subsystem, now consisting only of
positive roots; this is what we use as $\Delta_{\Psi}$.

$G$ | $V\downarrow\Bbbk[e]$ | Specification $\mathcal{S}(e)$ | Conditions
---|---|---|---
$A_{n}$ | $\sum^{r}_{i=1}V(k_{i})$ | $((k_{i})_{i=1}^{r})$ |  $\sum^{r}_{i=1}k_{i}=n+1$;
|  |  |  $(k_{i})_{i=1}^{r}$ decreasing
$B_{n}$ | $\sum^{r}_{i=1}W(k_{i})+\sum^{s}_{i=1}W_{l_{i}}(m_{i})$ | $((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},m)$ |  $\sum^{r}_{i=1}2k_{i}+\sum^{s}_{i=1}2m_{i}+2m-1=2n+1$;
| $+D(m)$ |  |  $(k_{i})_{i=1}^{r}$ decreasing;
|  |  |  $(m_{i})_{i=1}^{s}$, $(l_{i})_{i=1}^{s}$, $(m_{i}-l_{i})_{i=1}^{s}$ strictly decreasing;
|  |  |  $\frac{m_{i}+1}{2}<l_{i}\leq m_{i}$ and $m<l_{s}$
$C_{n}$ | $\sum^{r}_{i=1}W(k_{i})+\sum^{s}_{i=1}W_{l_{i}}(m_{i})$ | $((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},(n_{i})_{i=1}^{t})$ |  $\sum^{r}_{i=1}2k_{i}+\sum^{s}_{i=1}2m_{i}+\sum^{t}_{i=1}2n_{i}=2n$;
| $+\sum^{t}_{i=1}V(2n_{i})$ |  |  $(k_{i})_{i=1}^{r}$, $(n_{i})_{i=1}^{t}$ decreasing;
|  |  |  $(m_{i})_{i=1}^{s}$, $(l_{i})_{i=1}^{s}$, $(m_{i}-l_{i})_{i=1}^{s}$ strictly decreasing;
|  |  | multiplicity of any $n_{i}$ is at most two;
|  |  |  $0<l_{i}<\frac{m_{i}}{2}$;
|  |  | for all $i,j$ either $n_{j}>m_{i}-l_{i}$ or $n_{j}<l_{i}$
$D_{n}$ | $\sum^{r}_{i=1}W(k_{i})+\sum^{s}_{i=1}W_{l_{i}}(m_{i})$ | $((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s})$ |  $\sum^{r}_{i=1}2k_{i}+\sum^{s}_{i=1}2m_{i}=2n$;
|  |  |  $(k_{i})_{i=1}^{r}$ decreasing;
|  |  |  $(m_{i})_{i=1}^{s}$, $(l_{i})_{i=1}^{s}$, $(m_{i}-l_{i})_{i=1}^{s}$ strictly decreasing;
|  |  | $\frac{m_{i}+1}{2}<l_{i}\leq m_{i}$
Table 2. Standard forms and specifications of nilpotent orbits in classical
type

$G$ | $J_{1}$ | $J_{2}$ | $J^{\prime}_{2}$ | Overalgebra | Root definitions
---|---|---|---|---|---
$A_{n}$ | $\bigcup\limits_{i=1}^{r}\\{\alpha_{u_{i}+j}\\}_{j=1}^{k_{i}-1}$ |  |  | $A_{k_{1}-1}\ldots A_{k_{r}-1}$ |
$B_{n}$ | $\bigcup\limits_{i=1}^{r}\\{\alpha_{u_{i}+j}\\}_{j=1}^{k_{i}-1}$ |  |  | $A_{k_{1}-1}\ldots A_{k_{r}-1}$ | $\beta_{i}=\alpha_{v_{i+1}-2(m_{i}-l_{i})-1}+\cdots+\alpha_{v_{i+1}-1}+2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-1}+2\alpha_{n}$
| $\bigcup\limits_{i=1}^{s}\\{\alpha_{v_{i}+j}\\}_{j=1}^{m_{i}-1}$ | $\\{\beta_{i}\\}_{i=1}^{s}$ | $\\{\widetilde{\beta_{i}}\\}_{i=1}^{s}$ | $D_{m_{1}}\ldots D_{m_{s}}$ | $\widetilde{\beta_{i}}=\alpha_{v_{i+1}-1}+2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-1}+2\alpha_{n}$
| $\\{\alpha_{j}\\}_{j=n-m+2}^{n}$ |  |  | $B_{m-1}$ |
$C_{n}$ | $\bigcup\limits_{i=1}^{r}\\{\alpha_{u_{i}+j}\\}_{j=1}^{k_{i}-1}$ |  |  | $A_{k_{1}-1}\ldots A_{k_{r}-1}$ | $\beta_{i}=2\alpha_{v_{i}+l_{i}}+\cdots+2\alpha_{n-1}+\alpha_{n}$
| $\bigcup\limits_{i=1}^{s}\\{\alpha_{v_{i}+j}\\}_{j=1}^{m_{i}-1}$ | $\\{\beta_{i}\\}_{i=1}^{s}$ | $\\{\widetilde{\beta_{i}}\\}_{i=1}^{s}$ | $C_{m_{1}}\ldots C_{m_{s}}$ | $\widetilde{\beta_{i}}=2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-1}+\alpha_{n}$
| $\bigcup\limits_{i=1}^{t}\\{\alpha_{w_{i}+j}\\}_{j=1}^{n_{i}-1}$ | $\\{\gamma_{i}\\}_{i=1}^{t}$ | $\\{\gamma_{i}\\}_{i=1}^{t}$ | $C_{n_{1}}\ldots C_{n_{t}}$ | $\gamma_{i}=\begin{cases}2\alpha_{w_{i}+n_{i}}+2\alpha_{w_{i}+n_{i}+1}+\cdots+2\alpha_{n-1}+\alpha_{n}&\ \text{if }w_{i}+n_{i}<n,\\\ \alpha_{n}&\ \text{if }w_{i}+n_{i}=n.\\\ \end{cases}$
$D_{n}$ | $\bigcup\limits_{i=1}^{r}\\{\alpha_{u_{i}+j}\\}_{j=1}^{k_{i}-1}$ |  |  | $A_{k_{1}-1}\ldots A_{k_{r}-1}$ | $\beta_{i}=\begin{cases}\parbox{211.3883pt}{$\alpha_{v_{i+1}-2(m_{i}-l_{i})-1}+\cdots+\alpha_{v_{i+1}-1}$\\\ \phantom{m} $+2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}$}&\ \text{if }i<s,\\\ \alpha_{v_{i+1}-2(m_{i}-l_{i})}+\cdots+\alpha_{n}&\ \text{if }i=s.\\\ \end{cases}$
| $\bigcup\limits_{i=1}^{s}\\{\alpha_{v_{i}+j}\\}_{j=1}^{m_{i}-1}$ | $\\{\beta_{i}\\}_{i=1}^{s}$ | $\\{\widetilde{\beta_{i}}\\}_{i=1}^{s}$ | $D_{m_{1}}\ldots D_{m_{s}}$ | $\widetilde{\beta_{i}}=\begin{cases}\alpha_{v_{i+1}-1}+2\alpha_{v_{i+1}}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}&\ \text{ if }i<s,\\\ \alpha_{n}&\ \text{ if }i=s.\\\ \end{cases}$

where: $\displaystyle u_{i}=\sum_{j=1}^{i-1}k_{j}$ $\displaystyle\text{for
}i=1,\ldots,r+1,$ $\displaystyle v_{i}=u_{r+1}+\sum_{j=1}^{i-1}m_{j}$
$\displaystyle\text{for }i=1,\ldots,s+1,$ $\displaystyle
w_{i}=v_{s+1}+\sum_{j=1}^{i-1}n_{j}$ $\displaystyle\text{for }i=1,\ldots,t+1.$

######  Remark 2.2.

The Dynkin diagrams below should be of assistance in seeing the connection
between representatives and the specification $\mathcal{S}(e)$, as well as the
natural overalgebras.

Type $A$

[labels*=u_1 = 0, */.style=fill=white, draw=white]A1 [labels*=1,k_1-1,k_1 =
u_2,,,u_3,u_r,u_r+1,n, x/.style=root radius = 0.15cm]Ao.oxo.ox.xo.o (root 3)
circle (0.1cm); (root 6) circle (0.1cm); (root 7) circle (0.1cm); [A_k_1-1]12
[A_k_2-1]45 [A_k_r-1]89 [labels*=u_r+1 = n+1, */.style=fill=white,
draw=white]A1

Type $B$

[labels*=1,k_1-1,u_2,u_r+1 = v_1,,,,v_2,v_s,,,n, x/.style=root radius =
0.15cm]Bo.ox.xo.oox.xo.oo (root 3) circle (0.1cm); (root 4) circle (0.1cm);
(root 6) circle (0.1cm); (root 7) circle (0.1cm); (root 8) circle (0.1cm);
(root 9) circle (0.1cm); [A_k_1-1]12 [D_m_1]57 [B_m-1]1012 136northrightleft
136 o13 13~β_1

Type $D$

[labels*=1,k_1-1,u_2,u_r+1 = v_1,,,,v_2,v_s,,,n-1, x/.style=root radius =
0.15cm]Ao.ox.xo.oox.xo.oo (root 3) circle (0.1cm); (root 4) circle (0.1cm);
(root 6) circle (0.1cm); (root 7) circle (0.1cm); (root 8) circle (0.1cm);
(root 9) circle (0.1cm); [A_k_1-1]12 [D_m_1]57 [D_m_s]1012 136northrightleft
136 o13 13~β_1 1411northrightleft 1411 o14 14~β_s

######  Remark 2.3.

For type $D_{n}$, the specification
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s})$
determines the $\mathrm{O}(V)$-orbit of the nilpotent element
$e\in\operatorname{Lie}(\mathrm{SO}(V))$. By [LS12, Prop. 5.25], the class of
$e$ splits into two $\mathrm{SO}(V)$-orbits if and only if $s=0$ and all
$k_{i}$ are even. In that case, one sees that the representative given in
Table LABEL:tab:stdreps is contained in the standard Levi subalgebra of type
$A_{n-1}$ with base $1,\ldots,n-1$. For a representative in the other
$\mathrm{SO}(V)$-class, use $J^{\prime}$ instead of $J$, where $J^{\prime}$ is
formed from $J$ by replacing every occurrence of $\alpha_{n-1}$ with
$\alpha_{n}$.

######  Example 2.4 (Regular elements).

If $e$ is a regular element, say $e=\sum_{i=1}^{n}e_{i}$, then in types
$A_{n}$, $B_{n}$, $C_{n}$ and $D_{n}$ respectively, we have

$\mathcal{S}(e)=((n+1)),\quad((),(),(),n+1),\quad((),(),(),(n))\quad\text{ and
}\quad((),(n),(n)).$

Finally for this subsection, we take the opportunity to record a technical
lemma regarding the special roots defined in Table LABEL:tab:stdreps. The
proof is essentially by inspection and left to the reader.

######  Lemma 2.5.

Suppose $G$ has type $B$ or $D$. Then:

  1. (i)

the difference $\beta_{i}-\beta_{j}\not\in\Phi$ for $i,j\in\\{1,\ldots,s\\}$;

  2. (ii)

there exists $\gamma\in\Phi^{+}$ with $\beta_{j}-\gamma=\alpha_{i}$ if and
only if $l_{j}<m_{j}$ and $i=v_{j+1}-2(m_{j}-l_{j})$.

###  2.4. Standard forms and representatives for nilpotent classes for
exceptional groups

When $G$ is exceptional, there is no natural module available from which to
derive a classification of nilpotent orbits. Instead the Bala–Carter system of
labelling gives root systems of Levi subalgebras in which the element is
distinguished. The reader is invited to consult [LS04, §9] for an overview.
For the purposes of this paper, we need only the finite list of orbit
representatives from [Ste16], which are implemented in Magma (and GAP) and
available at https://github.com/davistem/nilpotent_orbits_magma.

###  2.5. Identification of orbits under embeddings

Given an embedding $\varphi:G\subseteq G^{\prime}$ of simple algebraic groups,
one gets a $G$-equivariant embedding
$\mathrm{d}\varphi:\mathcal{N}\to\mathcal{N}^{\prime}$ of nilpotent cones.
Typically, several $G$-orbits in $\mathcal{N}$ fuse into a single
$G^{\prime}$-orbit after applying $\mathrm{d}\varphi$. We spell out what
happens for two particular instances of $\varphi$.

$\varphi:\mathrm{O}_{2n+1}\to\mathrm{O}_{2n+2}$. Let $V$ be a non-degenerate
codimension-$1$ subspace of a $(2n+2)$-dimensional space $V_{0}$ on which a
non-singular quadratic form has been fixed. Then there is a natural embedding
$\varphi:\mathrm{O}(V)\to\mathrm{O}(V_{0})$. This embedding is used in [LS12,
Section 5.6] to describe the nilpotent cone of $\mathrm{O}(V)$. Let
$e\in\mathcal{N}(\mathrm{O}(V))$ with specification
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},m)$ and
suppose $e^{\prime}=\mathrm{d}\varphi(e)\in\mathcal{N}^{\prime}$. Define
$m_{s+1}=m$. Then
$\mathcal{S}(e^{\prime})=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s+1})$.

$\varphi:\mathrm{O}_{2m}\to\mathrm{Sp}_{2m}$. A non-degenerate quadratic form
on $V$ preserved by $\mathrm{O}(V)$ gives rise to a bilinear form on $V$,
which since $p=2$, is alternating. Thus we get an inclusion
$\varphi:\mathrm{O}(V)\subseteq\mathrm{Sp}(V)$. Indeed, an easy calculation
shows that $\operatorname{Lie}(\mathrm{O}(V))$ identifies with the derived
subalgebra of $\operatorname{Lie}(\mathrm{Sp}(V))$. Each orbit
$\mathcal{O}\subseteq\mathcal{N}$ is characterised by admitting a
representative $e\in\operatorname{Lie}(O(V))$ with

$V\downarrow\Bbbk[e]=\sum^{r}_{i=1}W(k_{i})+\sum^{s}_{i=1}W_{l_{i}}(m_{i}).$

In [LS12, 5.1(B)] we learn that $W_{l}(m)$ is a non-degenerate space of
dimension $2m$ which is a sum of two totally singular spaces of dimension $m$.
These two spaces are then also totally isotropic for the associated bilinear
form and so $e$ has the same class over $\mathrm{Sp}(V)$ as a nilpotent
element $e^{\prime}$, where

$V\downarrow\Bbbk[e^{\prime}]=\sum^{r}_{i=1}W(k_{i})+\sum^{s}_{i=1}W(m_{i}).$

(The reader should note that the two $W_{l}(m)$’s defined for $\mathrm{O}(V)$
and $\mathrm{Sp}(V)$ in [LS12, §5.1] have no relationship with each other.)
Also, observe that for any admissible sequence of $k_{i}$ such that there is
$e^{\prime}\in\operatorname{Lie}(\mathrm{Sp}(V))$ with
$V\downarrow\Bbbk[e]=\sum_{i=1}^{r}W(k_{i})$, then there is
$e\in\operatorname{Lie}(\mathrm{O}(V))$ with
$\mathrm{d}\varphi(e)=e^{\prime}$. Hence,

(5) $\text{if }e\in\mathfrak{sp}_{2n}\text{ with
}\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},(n_{i})_{i=1}^{t})\text{
then }e\in\mathfrak{so}_{2n}\Leftrightarrow s=t=0.$

##  3\. Solving $[h,e]=e$

In this section $G$ is a simple $\Bbbk$-group, classical or exceptional, and
$e$ is a nilpotent element of $\mathfrak{g}$. In pursuit of a classification
of the orbits $\mathcal{O}_{e}$ for which
$e\in\mathfrak{s}\subseteq\mathfrak{g}$ or
$e\in\mathfrak{pgl}_{2}\subseteq\mathfrak{g}$, we search first for those
$\mathcal{O}_{e}$ for which there is an element $h$ satisfying $[h,e]=e$. In
case $G$ is classical, recall the notation for the specifications
$\mathcal{S}(e)$ from Table 2 and the representatives from Table
LABEL:tab:stdreps.

######  Theorem 3.1.

There exists $h\in\mathfrak{g}$ such that $[h,e]=e$ if and only if at least
one of the following holds:

  1. (i)

$G$ is adjoint, or more generally $[X(T):\mathbb{Z}\Phi]$ is coprime to $2$;

  2. (ii)

$G\cong\mathrm{SO}_{2n}$;

  3. (iii)

$G$ is of type $A_{n}$ and either $n\not\equiv 1\pmod{4}$ or $\mathcal{S}(e)$
has at least one odd $k_{i}$;

  4. (iv)

$G\cong\mathrm{Spin}_{2n+1}$ and either $n\not\equiv 1,2\pmod{4}$ or
$\mathcal{S}(e)$ has at least one odd $k_{i}$ or $m_{i}$;

  5. (v)

$G\cong\mathrm{Sp}_{2n}$ and $\mathcal{S}(e)$ has $s=t=0$;

  6. (vi)

$G\cong\mathrm{Spin}_{2n}$ and either $n\not\equiv 2\pmod{4}$ or
$\mathcal{S}(e)$ has at least one odd $k_{i}$ or $m_{i}$;

  7. (vii)

$G\cong\mathrm{HSpin}_{4n}$, and

    1. (a)

$\mathcal{S}(e)$ has at least one odd $k_{i}$ or $m_{i}$;

    2. (b)

$\mathcal{S}(e)$ has $s=0$ and the standard representative of
$\mathcal{O}_{e}$ is supported on $e_{n}$;

  8. (viii)

$G$ is of type $E_{6}$;

  9. (ix)

$G$ is of type $E_{7}$ and $e$ has one of the following labels: $E_{6}$,
$E_{6}(a_{1})$, $A_{6}$, $D_{5}$, $E_{6}(a_{3})$, $(A_{5})^{\prime}$,
$A_{4}+A_{2}$, $D_{5}(a_{1})$, $A_{4}+A_{1}$, $A_{4}$, $A_{3}+A_{2}$, $D_{4}$,
$D_{4}(a_{1})$, $(A_{3}+A_{1})^{\prime}$, $2A_{2}+A_{1}$, $2A_{2}$, $A_{3}$,
$A_{2}+2A_{1}$, $A_{2}+A_{1}$, $A_{2}$, $(3A_{1})^{\prime}$, $2A_{1}$,
$A_{1}$.

######  Remark 3.2.

The condition on $\mathcal{S}(e)$ in part (v) can be restated as saying that
$e$ is contained in a Levi subalgebra of type $A_{n-1}$.

The rest of this section is concerned with the proof of Theorem 3.1. The
machine computation in the exceptional case will be discussed in Section 7.
Therefore we work under the following assumptions until further notice:

$G$ is classical and $e$ is a standard representative of $\mathfrak{g}$.

######  Lemma 3.3.

Theorem 3.1 holds when $[X(T):\mathbb{Z}\Phi]$ is coprime to $2$ or
$G\cong\mathrm{SO}_{2n}$.

###### Proof.

If $[X(T):\mathbb{Z}\Phi]$ is coprime to $2$, the isogeny from $G$ to $G/Z(G)$
is separable and so we may reduce to the adjoint case. Thus the matrix
$\mathsf{A}$ appearing in the factorisation
$\mathsf{C}=\mathsf{A}\mathsf{B}^{T}$ is the identity.

Let $h=\sum_{i=1}^{n}h_{i}$. Then since in all cases $e$ is a sum of root
vectors, we have $[h,e]=e$ by Equation (CB2). (For the case
$G\cong\mathrm{SO}_{2n}$ we refer to (3) in Section 1.) ∎

######  Lemma 3.4.

Suppose $G\cong\mathrm{Sp}_{2n}$. Then there exists $h\in\mathfrak{g}$ with
$[h,e]=e$ if and only if $e\in[\mathfrak{g},\mathfrak{g}]$. Hence Theorem 3.1
holds for $G$.

###### Proof.

If $[h,e]=e$, then clearly $e\in[\mathfrak{g},\mathfrak{g}]$. On the other
hand, $\mathfrak{m}:=[\mathfrak{g},\mathfrak{g}]=\operatorname{Lie}(M)$ for a
closed subgroup $M\cong\mathrm{SO}_{2n}$ and so if $e\in\mathfrak{m}$ then
Lemma 3.3 gives the result. The characterisation of $e$ in Theorem 3.1 follows
from (5). ∎

Lemmas 3.3 and 3.4 deal completely with type $C$. So we may assume from now on
that

$G$ has type $A$, $B$ or $D$.

######  Lemma 3.5.

Suppose there exists $h\in\mathfrak{g}$ such that $[h,e]=e$. Then there is
some $h_{0}\in\mathfrak{h}$ such that $[h_{0},e]=\pi_{1}(e)$.

###### Proof.

Write $h=h_{-}+h_{0}+h_{+}$ where $h_{-}\in\mathfrak{g}_{-}$,
$h_{+}\in\mathfrak{g}_{+}$ and $h_{0}\in\mathfrak{g}_{0}=\mathfrak{h}$.
(Recall the notation from Section 2.1.)

Suppose $G$ is of type $A_{n}$. Since $e$ is a standard representative, it is
a sum of simple root vectors and so $e\in\mathfrak{g}_{1}$. Then
$[h_{-},e]\subseteq\mathfrak{g}_{-}$, $[h_{+},e]\subseteq\mathfrak{g}_{\geq
2}$ and $[h_{0},e]\subseteq\mathfrak{g}_{1}$. Therefore, $[h,e]=[h_{0},e]=e$.

Now let $G$ be of type $B_{n}$ and suppose the statement of the theorem does
not hold. Then $[h_{0},e]\neq\pi_{1}(e)$, which implies there is
$\alpha_{i}\in J_{1}$ with $[h_{0},e_{i}]\neq e_{i}$.

Since $\pi_{1}([h_{+},e])=0$, we must have $[h_{-},e]$ supported on $e_{i}$.
Recall that $e$ is a sum of root vectors for roots of height $1$ or roots of
the form $\beta_{j}\in J_{2}$ (Table LABEL:tab:stdreps). As
$\operatorname{ad}(h_{-})$ takes $\mathfrak{g}_{1}$ to $\mathfrak{g}_{-}$,
there must be $\beta_{j}\in J_{2}$ with $[h_{-},e_{\beta_{j}}]$ supported on
$e_{i}$. So $h_{-}$ is supported on $e_{-\gamma_{1}}$ for some
$\gamma_{1}\in\Phi^{+}$ with $\beta_{j}-\gamma_{1}=\alpha_{i}$. Lemma 2.5(ii)
shows that $m_{j}\neq l_{j}$ and $i=v_{j+1}-2(m_{j}-l_{j})$ and hence

$\displaystyle\gamma_{1}=$
$\displaystyle(\beta_{j}-\alpha_{v_{j+1}-2(m_{j}-l_{j})})$ $\displaystyle=$
$\displaystyle-\left(\alpha_{v_{j+1}-2(m_{j}-l_{j})+1}+\cdots+\alpha_{v_{j+1}}+2\alpha_{v_{j+1}+1}+\cdots+2\alpha_{n-2}+2\alpha_{n-1}+2\alpha_{n}\right).$

We know $[h,e]=e$ and that $[h_{0}+h_{+},e]$ is only supported on positive
roots. Thus $[h_{-},e]$ cannot be supported on negative roots. However,
observe that $e$ is supported on $e_{v_{j+1}-2(m_{j}-l_{j})+1}$ and $h_{-}$ is
supported on $e_{-\gamma_{1}}$. Now,
$[e_{-\gamma_{1}},e_{v_{j+1}-2(m_{j}-l_{j})+1}]=e_{\delta}$ where

$\displaystyle\delta=$
$\displaystyle-\gamma_{1}+\alpha_{v_{j+1}-2(m_{j}-l_{j})+1}$ $\displaystyle=$
$\displaystyle-\left(\alpha_{v_{j+1}-2(m_{j}-l_{j})+2}+\cdots+\alpha_{v_{j+1}}+2\alpha_{v_{j+1}+1}+\cdots+2\alpha_{n-2}+2\alpha_{n-1}+2\alpha_{n}\right)\in\Phi^{-}.$

Since $[h_{-},e]$ is not supported on $\delta$, there must also exist
$\gamma_{2}\in\Phi^{+}\setminus\\{\gamma_{1}\\}$ such that
$\delta=\zeta-\gamma_{2}$ for some $\zeta\in J$, and such that $h$ is
supported on $-\gamma_{2}$. Inspection of $\delta$ immediately shows that
$\zeta\not\in J_{2}$. Therefore $\zeta\in J_{1}$ and $\zeta$ is simple. We
have $\delta-\zeta$ is a root, and so
$\zeta=\alpha_{v_{j+1}-2(m_{j}-l_{j})+1}$ or $\zeta=\alpha_{v_{j+1}}$. The
latter is not in $J$ and the former yields $\gamma_{2}=\gamma_{1}$, a
contradiction.

When $G$ is of type $D$ the proof follows exactly the same way as for $B$.∎

######  Lemma 3.6.

Theorem 3.1 holds when $G$ is of type $A$.

###### Proof.

By Lemma 3.3 we may assume $[X(T):\mathbb{Z}\Phi]$ is divisible by $2$. Let
$\mathcal{S}(e)=((k_{i})_{1\leq i\leq r})$. We start by exhibiting solutions
to $[h,e]=e]$ in case $n\not\equiv 1\pmod{4}$ or $\mathcal{S}(e)$ has an odd
$k_{i}$. It suffices to treat the case $G=\mathrm{SL}_{n+1}$, since if we can
solve $[h,e]=e$ in $\mathfrak{g}$, we can solve it in any central quotient.
Recall

$e=\sum_{i=1}^{r}\sum_{j=1}^{k_{i}-1}e_{u_{i}+j}$

is a standard representative. We define

$h=\sum_{i=0}^{\lfloor\frac{n-2}{4}\rfloor}(h_{4i+1}+h_{4i+2})\quad\text{ for
}n\equiv 2,3\pmod{4},\quad
h=\sum_{i=0}^{\frac{n}{4}}(h_{4i+2}+h_{4i+3})\quad\text{ for }n\equiv
0\pmod{4}.$

It is routine to use (CB2) to check that $[h,e]=e$.

Now assume $n\equiv 1\pmod{4}$ and $\mathcal{S}(e)$ has an odd $k_{i}$. Then
we may write $e=e^{\prime}+e^{\prime\prime}$, where $e^{\prime}$ and
$e^{\prime\prime}$ are contained in commuting subalgebras
$\mathfrak{g}^{\prime}$ and $\mathfrak{g}^{\prime\prime}$ of type
$A_{k_{i}-1}$ and $A_{n-k_{i}}$, respectively. Both $k_{i}-1$ and $n-k_{i}$
are even, so by the previous paragraph, we have $h^{\prime}$ and
$h^{\prime\prime}$ such that $[h^{\prime},e^{\prime}]=e^{\prime}$ and
$[h^{\prime\prime},e^{\prime\prime}]=e^{\prime\prime}$. Then
$h:=h^{\prime}+h^{\prime\prime}$ satisfies $[h,e]=e$.

On the other hand, consider the case that $n\equiv 1\pmod{4}$ and
$\mathcal{S}(e)$ has no odd $k_{i}$. Suppose for a contradiction that
$[h,e]=e$. Since $n+1\equiv 2\pmod{4}$, there is a separable isogeny from
$\mathrm{SL}_{n+1}$ to $G$ which differentiates to an isomorphism of Lie
algebras, and so it suffices to treat the case $G\cong\mathrm{SL}_{n+1}$. With
no odd $k_{i}$ and $n\equiv 1\pmod{4}$, there must be an odd number of $i$ for
which $k_{i}\equiv 2\pmod{4}$ and an even number of $i$ for which $k_{i}\equiv
0\pmod{4}$. In particular, $r$ is odd. By Lemma 3.5, there is a solution to
$[h_{0},e]=e$ for $h_{0}\in\mathfrak{h}$, so let
$h_{0}=\sum_{i}\lambda_{i}h_{i}$ be a generic element of $\mathfrak{h}$.

The formulas in Section 1 yield

$[h_{0},e]=\sum_{i=1}^{r}\sum_{j=1}^{k_{i}-1}(\lambda_{u_{i}+j-1}+\lambda_{u_{i}+j+1})e_{u_{i}+j},$

where we set $\lambda_{0}=\lambda_{n+1}=0$. Since $[h_{0},e]=e$ for each $i$
we must have $\lambda_{u_{i}+j-1}+\lambda_{u_{i}+j+1}=1$ for all
$j\in\\{1,\ldots,k_{i}-1\\}$. Every $k_{i}$ is even, so it follows that
$\lambda_{2j}+\lambda_{2j+2}=1$ for all $j\in\\{0,\ldots,\frac{n-1}{2}\\}$.
But as $n\equiv 1\pmod{4}$ we obtain the contradiction

$1=\sum_{j=0}^{\frac{n-1}{2}}(\lambda_{2j}+\lambda_{2j+2})=\lambda_{0}+\lambda_{n+1}=0+0=0.\qed$

######  Lemma 3.7.

Theorem 3.1 holds when $G$ is of type $D$.

###### Proof.

By Lemma 3.3 we can assume $G\cong\mathrm{Spin}_{2n}$ or
$G\cong\mathrm{HSpin}_{2n}$. Suppose $n\not\equiv 2\pmod{4}$. We exhibit a
solution to $[h,e]=e$. It suffices to treat the case
$G\cong\mathrm{Spin}_{2n}$, since this will imply a solution in
$\mathrm{HSpin}_{2n}$. Let

$h=\sum_{i=0}^{\lfloor\frac{n-3}{4}\rfloor}(h_{4i+1}+h_{4i+2})\quad\text{ if
}n\equiv 0,3\pmod{4},\quad
h=\sum_{i=0}^{\lfloor\frac{n-2}{4}\rfloor}(h_{4i+2}+h_{4i+3})\quad\text{ if
}n\equiv 1\pmod{4}.$

Using (CB2) (the relevant matrix $\mathsf{A}$ is the Cartan matrix of $G$) we
find that if $\gamma=\sum_{i}c_{i}\alpha_{i}$ then
$[h,e_{\gamma}]=(\sum_{i}c_{i})e_{\gamma}=\text{ht}(\gamma)e_{\gamma}$. But
the height of all the roots in $J$ is seen to be odd. It therefore follows
that $[h,e]=e$.

Now let $n\equiv 2\pmod{4}$ and
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s})$.
First, suppose there is an odd $k_{i}$ or $m_{i}$ and let this odd number be
$x$. Then $e=e^{\prime}+e^{\prime\prime}$ with $e^{\prime}$ and
$e^{\prime\prime}$ being standard nilpotent elements in commuting subalgebras
$\mathfrak{g}^{\prime}$ and $\mathfrak{g}^{\prime\prime}$ of type $D_{x}$ and
$D_{n-x}$, respectively. Since both $x$ and $n-x$ are odd, we have established
the existence of solutions to $[h^{\prime},e^{\prime}]=e^{\prime}$ and
$[h^{\prime\prime},e^{\prime\prime}]=e^{\prime\prime}$ in
$\mathfrak{g}^{\prime}$ and $\mathfrak{g}^{\prime\prime}$. Thus we get the
solution $h:=h^{\prime}+h^{\prime\prime}$ to $[h,e]=e$.

Now we assume all $k_{i}$ and $m_{i}$ are even. As all $k_{i}$ and $m_{i}$ are
even and $n\equiv 2\pmod{4}$, it follows that $\alpha_{2j-1}\in J$ for each
$j=1,\ldots,\frac{n}{2}-1$; and at least one of $\alpha_{n-1}$ and
$\alpha_{n}$ is also in $J$. Recall that a solution to $[h,e]=e$ yields an
element $h_{0}\in\mathfrak{h}$ such that $[h_{0},e]=\pi_{1}(e)$ by Lemma 3.5.
Let $h_{0}=\sum_{i}\lambda_{i}h_{i}$ be a general element of $\mathfrak{h}$.
Since $h_{0}\in\mathfrak{h}$ and $e_{2j-1}\in\mathfrak{g}_{1}$ we must have
$[h_{0},e_{2j-1}]=e_{2j-1}$ for each such $j=1,\ldots,\frac{n}{2}-1$.

Now take the case $G=\mathrm{Spin}_{2n}$. Using (CB2) we see that
$[h_{0},e_{2j-1}]=(\lambda_{2j-2}+\lambda_{2j})e_{2j-1}$ for
$j=2,\ldots,\frac{n}{2}-1$, $[h_{0},e_{1}]=\lambda_{2}e_{1}$,
$[h_{0},e_{n-1}]=\lambda_{n-2}e_{n-1}$ and $[h_{0},e_{n}]=\lambda_{n-2}e_{n}$.
Therefore $(\lambda_{2j-2}+\lambda_{2j})=1$ for $j=1,\ldots,\frac{n}{2}-1$ and
$\lambda_{2}=\lambda_{n-2}=1$. However, $n\equiv 2\pmod{4}$ so

$1=\sum_{j=2}^{\frac{n}{2}-1}(\lambda_{2j-2}+\lambda_{2j})=\lambda_{2}+\lambda_{n-2}=1+1=0.$

Thus there is no solution to $[h,e]=e$ when all $k_{i}$ and $m_{j}$ are even
in the $\mathrm{Spin}_{2n}$ case.

Finally, suppose $G\cong\mathrm{HSpin}_{2n}$. If $s=0$ then there are two
classes of nilpotent elements in $\text{Lie}(G)$, which can be distinguished
by which Levi subalgebra $\mathfrak{l}$ of type $A_{n-1}$ they are contained
in; see Remark 2.3. If $e$ is contained in the Levi with simple roots
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n}$, then there is a solution to
$[h,e]=e$. Indeed, $h_{0}=h_{1}+\cdots+h_{n-2}+h_{n}$ works since
$[h_{i},e_{i}]=e_{i}$ for $i=1,\ldots,n-2,n$ by Equation (4).

For all other classes (with $s=0$ or otherwise) $e$ is supported on $e_{n-1}$.
Using Equation (4) we see
$[h_{0},e_{2j-1}]=(\lambda_{2j-1}+\lambda_{n})e_{2j-1}$ and so
$\lambda_{2j-1}+\lambda_{n}=1$ for $j=1,\ldots,\frac{n}{2}-1$. Also,
$[h_{0},e_{n-1}]=(\sum_{j=1}^{\frac{n}{2}-1}\lambda_{2j-1})e_{n-1}$ so
$(\sum_{j=1}^{\frac{n}{2}-1}\lambda_{2j-1})$ must also be $1$ if there is to
be a solution. However, $n\equiv 2\pmod{4}$ so $(\frac{n}{2}-1)\equiv
0\pmod{4}$ and thus

$0=\sum_{j=1}^{\frac{n}{2}-1}1=\sum_{j=1}^{\frac{n}{2}-1}(\lambda_{2j-1}+\lambda_{n})=\left(\sum_{j=1}^{\frac{n}{2}-1}\lambda_{2j-1}\right)+\left(\frac{n}{2}-1\right)\lambda_{n}=\left(\sum_{j=1}^{\frac{n}{2}-1}\lambda_{2j-1}\right).$

Therefore, there is no solution to $[h_{0},e]=\pi_{1}(e)$ in this case. ∎

######  Lemma 3.8.

Theorem 3.1 holds for $G\cong\mathrm{Spin}_{2n+1}$.

###### Proof.

Let $e$ have specification
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s},m)$.
Recall from Section 2.5 the embedding of $G$ into
$M\cong\mathrm{Spin}_{2n+2}$. Then $e$ is conjugate to an element $e^{\prime}$
of $\mathfrak{m}=\operatorname{Lie}(M)$ with
$\mathcal{S}(e^{\prime})=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s+1})$
for $m_{s+1}=m$.

If $n\equiv 0$ or $3\pmod{4}$, then the rank of $M$ is $1$ or $0\pmod{4}$,
respectively, and take $h\in\mathfrak{m}$ with $[h,e]=e$ from the proof of
Lemma 3.7. This $h$ is visibly stable under the standard graph automorphism of
$M$ and therefore we have $h\in\mathfrak{g}$.

When $n\equiv 1\pmod{4}$ and all $k_{i}$, $m_{j}$ are even it follows that $m$
is also even. Thus Lemma 3.7 shows that there is no solution to
$[h,e^{\prime}]=e^{\prime}$ in $\mathfrak{m}$ and hence no solution to
$[h,e]=e$ in $\mathfrak{g}$. If $n\equiv 2\pmod{4}$, the requirement for at
least one $k_{i}$ or $m_{j}$ to be odd follows using a very similar argument
to that in the proof of Lemma 3.7.

For the remainder of the proof $n\equiv 1,2\pmod{4}$ and an odd $k_{i}$ or
$m_{i}$ exists, which we call $x$. It follows from the explanations in Section
2.3 that $e$ is contained in a subsystem subalgebra
$\mathfrak{m}_{1}+\mathfrak{m}_{2}$ with $\mathfrak{m}_{1}$ of type $D_{x}$
and $\mathfrak{m}_{2}$ of type $B_{n-x}$. Furthermore, we may assume that
$e=y_{1}+y_{2}$ with $y_{i}\in\mathfrak{m}_{i}$ nilpotent elements in standard
form. For ease, we choose bases
$\mathcal{B}_{1}=\\{\alpha_{1},\ldots,\alpha_{x-1},\alpha_{x-1}+2\alpha_{x}+\cdots+2\alpha_{n}\\}$
(when $x>1$) and $\mathcal{B}_{2}=\\{\alpha_{x+1},\ldots,\alpha_{n}\\}$ for
the subsystems $D_{x}$ and $B_{n-x}$, respectively. We proceed by finding
$h\in\mathfrak{g}$ with $[h,y_{1}]=y_{1}$ and $[h,y_{2}]=y_{2}$ for any
standard nilpotent elements $y_{i}\in\mathfrak{m}_{i}$. Lemma 3.7 and the
previous cases in this proof show the existence of $h_{i}\in\mathfrak{m}_{i}$
with $[h_{i},y_{i}]=y_{i}$ whenever $n-x\equiv 0,3\pmod{4}$. Thus the sum
$h=h_{1}+h_{2}$ is sufficient unless $(n,x)\equiv(1,3)$ or $(2,1)\pmod{4}$.

For these two cases let

$h=\sum\limits_{i=0}^{\frac{x-7}{4}}(h_{4i+2}+h_{4i+3})+h_{x-1}+h_{x}+h_{x+1}+\sum\limits_{i=0}^{\frac{n-x-6}{4}}(h_{4i+x+4}+h_{4i+x+5}),$

if $(n,x)\equiv(1,3)\pmod{4}$ and

$h=\sum\limits_{i=0}^{\frac{x-5}{4}}(h_{4i+1}+h_{4i+2})+h_{x}+\sum\limits_{i=0}^{\frac{n-x-5}{4}}(h_{4i+x+3}+h_{4i+x+4}),$

if $(n,x)\equiv(2,1)\pmod{4}$.

We claim $[h,y_{i}]=y_{i}$. To see this, first use (CB2) to see that if
$\delta=\sum_{i}c_{i}\alpha_{i}$ is any root, then

(6) $[h,e_{\delta}]=\sum_{i\neq x}c_{i}e_{\delta}.$

Therefore, $[h,e_{\alpha}]=e_{\alpha}$ for $\alpha\in\mathcal{B}_{i}$.
Inspecting the definitions of the roots $\beta_{i}\in J_{2}$ in Table
LABEL:tab:stdreps for types $B$ and $D$, we see $[h,e_{\beta}]=e_{\beta}$ for
the relevant additional roots $\beta$ in our standard nilpotent elements
$y_{1}\in\mathfrak{m}_{1}$ and $y_{2}\in\mathfrak{m}_{2}$, thus completing the
proof. ∎

##  4\. Extending nilpotent elements to $\mathfrak{s}$-subalgebras

The simple $\Bbbk$-group $G$ reverts to its general state.

It is a straightforward exercise to check that there is a unique simple Lie
algebra $\mathfrak{s}$ of dimension $3$. It has basis $\\{e,h,f\\}$ subject to
the relations

(7) $[e,f]=h,\ [h,e]=e,\ [h,f]=f.$

######  Theorem 4.1.

Suppose $e$ is a non-zero nilpotent element of $\mathfrak{g}$. Then $e$ is
contained in a subalgebra $\mathfrak{s}$ if and only if

  1. (i)

$G$ is of type $A_{n}$ and $\mathcal{S}(e)$ has no $k_{i}\equiv 2\pmod{4}$;

  2. (ii)

$G$ is of type $B_{n}$ and $\mathcal{S}(e)$ has $m\not\equiv 2,3\pmod{4}$ and
no $k_{i},m_{i}\equiv 2\pmod{4}$;

  3. (iii)

$G$ is of type $C_{n}$ and $\mathcal{S}(e)$ has $s=t=0$ and no $k_{i}\equiv
2\pmod{4}$;

  4. (iv)

$G$ is of type $D_{n}$ and $\mathcal{S}(e)$ has no $k_{i},m_{i}\equiv
2\pmod{4}$;

  5. (v)

$G$ is of type $G_{2}$ and $e$ has label: $G_{2}$, $G_{2}(a_{1})$;

  6. (vi)

$G$ is of type $F_{4}$ and $e$ has label: $F_{4}$, $F_{4}(a_{1})$,
$F_{4}(a_{2})$, $B_{3}$, $F_{4}(a_{3})$, $(B_{2})^{(2)}$, $\tilde{A}_{2}$,
$A_{2}$;

  7. (vii)

$G$ is of type $E_{6}$ and $e$ has label: $E_{6}$, $E_{6}(a_{1})$, $D_{5}$,
$E_{6}(a_{3})$, $D_{5}(a_{1})$, $D_{4}$, $A_{4}$, $D_{4}(a_{1})$, $A_{3}$,
$2A_{2}$, $A_{2}$;

  8. (viii)

$G$ is of type $E_{7}$ and $e$ has label: $E_{6}$, $E_{6}(a_{1})$, $A_{6}$,
$D_{5}$, $E_{6}(a_{3})$, $A_{4}+A_{2}$, $D_{5}(a_{1})$, $A_{4}$,
$A_{3}+A_{2}$, $D_{4}$, $D_{4}(a_{1})$, $2A_{2}$, $A_{3}$, $A_{2}$;

  9. (ix)

$G$ is of type $E_{8}$ and $e$ has label: $E_{8}$, $E_{8}(a_{1})$,
$E_{8}(a_{2})$, $E_{8}(a_{3})$, $E_{8}(a_{4})$, $E_{8}(a_{5})$,
$E_{8}(b_{5})$, $(D_{7})^{(2)}$, $D_{7}$, $E_{8}(a_{6})$, $D_{7}(a_{1})$,
$E_{8}(b_{6})$, $A_{7}$, $D_{7}(a_{2})$, $E_{6}$, $D_{5}+A_{2}$,
$E_{6}(a_{1})$, $A_{6}$, $E_{8}(a_{7})$, $D_{5}(a_{1})+A_{2}$, $A_{4}+A_{3}$,
$D_{5}$, $E_{6}(a_{3})$, $D_{4}+A_{2}$, $A_{4}+A_{2}$, $D_{5}(a_{1})$,
$2A_{3}$, $A_{4}$, $A_{3}+A_{2}$, $D_{4}$, $D_{4}(a_{1})$, $2A_{2}$, $A_{3}$,
$A_{2}$.

We start with some preliminary results about $\mathfrak{s}$. Since
$\mathfrak{s}$ is simple, it has a unique minimal $p$-envelope $\mathfrak{S}$,
and there is a unique $[p]$-map on $\mathfrak{S}$, [SF88, p97]. Computing with
an adjoint basis, we have $h^{[2]}=h$, $e^{[2]}=E$ and $f^{[2]}=F$, where
$[E,e]=[E,h]=[F,h]=[F,f]=0$, $[E,f]=e$ and $[F,e]=f$. Furthermore
$E^{[2]}=F^{[2]}=0$. Thus $\\{E,e,h,f,F\\}$ is a basis of $\mathfrak{S}$ (with
a root system of type $BC_{1}$ according to an appropriate $1$-torus
$T\subseteq\operatorname{Aut}\mathfrak{s}$).

######  Lemma 4.2.

The nilpotent cone $\mathcal{N}(\mathfrak{S})$ of $\mathfrak{S}$ is the
variety

$\\{\lambda E+\mu e+\nu h+\rho f+\sigma
F\mid\lambda\rho=\mu\nu,\mu\sigma=\nu\rho\\}.$

We have
$\mathcal{N}(\mathfrak{s}):=\mathcal{N}(\mathfrak{S})\cap\mathfrak{s}=\Bbbk\cdot
e\oplus\Bbbk\cdot f$ as a subvariety. Furthermore, $\mathcal{N}(\mathfrak{s})$
identifies with the variety $\mathcal{M}:=\\{n\in\mathfrak{s}\mid
n\in\operatorname{im}(\operatorname{ad}n)^{2}\\}$.

###### Proof.

If $n=\lambda E+\mu e+\nu h+\rho f+\sigma F$, then according to the basis
$\\{e,f,h\\}$, the element $N:=\operatorname{ad}n$ is represented by the
matrix

$\begin{bmatrix}\nu&\mu&\lambda\\\ \rho&0&\mu\\\
\sigma&\rho&\nu\end{bmatrix},$

with characteristic polynomial
$\chi_{N}(t)=t^{3}+(\nu^{2}+\lambda\sigma)t+\lambda\rho^{2}+\sigma\mu^{2}$.
Now, $n$ is nilpotent if and only if $\chi_{N}(t)=t^{3}$, from which the
descriptions of $\mathcal{N}(\mathfrak{S})$ and $\mathcal{N}(\mathfrak{s})$
follow. If $n\in\operatorname{im}\operatorname{ad}(n)^{2}$ then
$n\in\mathfrak{s}$; so setting $\lambda=\sigma=0$ and squaring $N$ we get

$N^{2}=\begin{bmatrix}\nu^{2}+\mu\rho&\nu\mu&\mu^{2}\\\ \nu\rho&0&\nu\mu\\\
\rho^{2}&\nu\rho&\nu^{2}+\mu\rho\end{bmatrix}.$

It is easily checked that $\mathcal{N}(\mathfrak{s})\subseteq\mathcal{M}$. On
the other hand, if there is a vector $v=[a,b,c]^{T}$ with
$N^{2}v=[\mu,\nu,\rho]^{T}$ then resolving along the second coordinate yields
the equation $a\nu\rho+c\nu\mu=\nu$. If $\nu\neq 0$ then we have
$a\rho+c\mu=1$ and this leads to a contradiction with the other two
equations.∎

The existence of a non-zero element $n\in\mathcal{M}$ is actually enough to
distinguish $\mathfrak{s}$ among $3$-dimensional Lie algebras.

######  Lemma 4.3.

Let $L$ be a Lie algebra over $\Bbbk$ of dimension at most $3$. If there
exists $0\neq e\in L$ such that
$e\in\operatorname{im}(\operatorname{ad}e)^{2}$ then $L\cong\mathfrak{s}$.

###### Proof.

Let $f\in L$ satisfy $[e,[e,f]]=e$ and define $h=[e,f]$. Then we claim that
$e,f,h$ is a basis for $L$. Clearly $e$ and $f$ are linearly independent.
Suppose that $h=\lambda e+\mu f$. Then $e=[e,h]=\mu h$ which implies that
$\mu\lambda=1$ and $\mu^{2}=0$, a contradiction which proves the claim.

Now,

$[e,[h,f]]=[[e,h],f]+[h,[e,f]]=[e,f]+[h,h]=[e,f].$

So $[h,f]+f\in\mathfrak{c}_{L}(e)$. To calculate $\mathfrak{c}_{L}(e)$,
suppose $x=ae+bf+ch\in\mathfrak{c}_{L}(e)$. Then $[x,e]=bh+ce$, so $b=c=0$ and
thus $\mathfrak{c}_{L}(e)=\langle e\rangle$. Hence $[h,f]=f+\lambda e$.

Setting $e^{\prime}=e$, $h^{\prime}=\sqrt{\lambda}e+h$ and
$f^{\prime}=\sqrt{\lambda}h+f$, we see that
$(e^{\prime},h^{\prime},f^{\prime})$ satisfy the relations (7).∎

If $\mathfrak{s}\subseteq\mathfrak{g}$, then being simple, $\mathfrak{s}$
survives in any central quotient
$\pi:\mathfrak{g}\to\mathfrak{g}/\mathfrak{z}$. On the other hand if
$\mathfrak{s}\subseteq\mathfrak{g}/\mathfrak{z}$, then $\mathfrak{s}$ lifts to
a subalgebra $\mathfrak{s}^{\prime}\cong\mathfrak{s}$ provided
$0\to\mathfrak{z}\to\pi^{-1}(\mathfrak{s})\to\mathfrak{s}\to 0$ splits. The
next result, classifying the central extensions of $\mathfrak{s}$, guarantees
this.

######  Lemma 4.4.

We have $H^{2}(\mathfrak{s},\Bbbk)=0$.

###### Proof.

We use [Wei94, Cor. 7.7.3]. Considering $\Bbbk$ as a left
$\mathfrak{s}$-module, the cohomology modules
$H^{\bullet}(\mathfrak{s},\Bbbk)$ are the cohomology of the cochain complex
$\mathrm{Hom}_{k}(\bigwedge^{\bullet}(\mathfrak{s}),\Bbbk)\cong(\bigwedge^{\bullet}(\mathfrak{s}))^{*}$.

For $a_{1},a_{2}\in\mathfrak{s}$ and $x\in\\{e,f,h\\}$ we calculate

$\delta_{1}x^{*}(a_{1},a_{2})=a_{1}x^{*}(a_{2})+a_{2}x^{*}(a_{1})+x^{*}([a_{1},a_{2}])$
$=x^{*}([a_{1},a_{2}]),$

since the action of $\mathfrak{s}$ is trivial on $\Bbbk$. Thus,
$\delta_{1}e^{*}(e,f)=0,\delta_{1}e^{*}(e,h)=1$ and $\delta_{1}e^{*}(f,h)=0$,
so $\delta_{1}e^{*}=(e,h)^{*}$. Similarly, $\delta_{1}f^{*}=(f,h)^{*}$ and
$\delta_{1}h^{*}=(e,f)^{*}$. We now see that the image of $\delta_{1}$ spans
the $3$-dimensional space $\bigwedge^{2}(\mathfrak{s})^{*}$ and hence
$H^{2}(\mathfrak{s},\Bbbk)=0$. ∎

Using the above lemma we may reduce to the situation where $G$ is simply
connected. We leave the discussion of computations when $G$ is of exceptional
type to Section 7. Hence, for the remainder of this section, we assume:

$G\cong\mathrm{SL}_{n}$, $\mathrm{Spin}_{2n+1}$, $\mathrm{Sp}_{2n}$ or
$\mathrm{Spin}_{2n}$ and $e$ is a representative from Table LABEL:tab:stdreps.

We continue with a useful observation.

######  Lemma 4.5.

Suppose $e=\sum_{j\in J_{1}}e_{j}$ is a sum of simple root vectors. Then
$e\in\operatorname{im}(\operatorname{ad}e)^{2}$ if and only if there is a
solution to $(\operatorname{ad}e)^{2}(\sum_{j\in J_{1}}a_{i}e_{-j})=e$.

###### Proof.

One direction is trivial. For the other, let $f\in\mathfrak{g}$ be such that
$(\operatorname{ad}e)^{2}f=e$. Since $e\in\mathfrak{g}_{1}$ we have
$e=\pi_{1}((\operatorname{ad}e)^{2}f)=\pi_{1}((\operatorname{ad}e)^{2}(\pi_{-1}(f)))=(\operatorname{ad}e)^{2}(\pi_{-1}(f))$.
We may write $\pi_{-1}(f)=\sum_{i=1}^{n}a_{i}e_{-i}$ for some $a_{i}$. Since
$[e_{j},e_{-i}]=0$ for $i\neq j$, we get $[e,\pi_{-1}(f)]=\sum_{j\in
J}a_{j}h_{j}$ as required. ∎

######  Lemma 4.6.

Theorem 4.1 holds for $G=\mathrm{SL}_{n+1}$.

###### Proof.

Suppose $\mathcal{S}(e)=((k_{i})_{1\leq i\leq r})$. Then $e$ is a sum of
regular nilpotent elements in a subsystem subalgebra $\mathfrak{m}$ which is a
commuting sum $\mathfrak{m}_{1}+\dots+\mathfrak{m}_{r}$ with each
$\mathfrak{m}_{i}$ of type $A_{k_{i}-1}$ (Table LABEL:tab:stdreps). If
$e\in\mathfrak{s}$ then there is $f$ such that
$(\operatorname{ad}e)^{2}(f)=e$. By Lemma 4.5, we may assume
$f=\sum_{\alpha\in J}a_{\alpha}e_{-\alpha}$, an element of $\mathfrak{m}$.
Such an $f$ exists if and only if it does on projection to each factor
$\mathfrak{m}_{i}$ of $\mathfrak{m}$. Hence it suffices to treat the case
$r=1$ and $e$ is regular.

Theorem 3.1(iii) immediately tells us that there is no $\mathfrak{s}$
overalgebra of $e$ when $n\equiv 1\pmod{4}$ since _a fortiori_ there is no
element $h$ such that $[h,e]=e$. Now suppose $n\not\equiv 1\pmod{4}$. Let

$f=\sum_{i=0}^{\lfloor\frac{n-2}{4}\rfloor}f_{4i+1}+f_{4i+2}\quad\text{ for
}n\equiv 2,3\pmod{4},\quad
f=\sum_{i=0}^{\frac{n}{4}}f_{4i+2}+f_{4i+3}\quad\text{ for }n\equiv
0\pmod{4}.$

Using $[e_{i},f_{j}]=\delta_{ij}h_{i}$, we see $h:=[e,f]$ is the same as that
specified in the proof of Theorem 3.6; in particular, $[e,[e,f]]=e$. It is
another easy calculation to see $[h,f]=f$, which gives $\langle
e,h,f\rangle\cong\mathfrak{s}$. ∎

The type $C$ case now follows easily.

######  Lemma 4.7.

Theorem 4.1 holds when $G\cong\mathrm{Sp}_{2n}$.

###### Proof.

Suppose $\langle e,h,f\rangle\cong\mathfrak{s}\subseteq\mathfrak{g}$ with
$[h,e]=e$. Theorem 3.1 yields $\mathcal{S}(e)=((k_{i})_{i=1}^{r},(),(),())$.
By Table LABEL:tab:stdreps, this means $e=\sum_{j\in J_{1}}e_{j}$ is a sum of
simple root vectors, such that $\alpha_{n}\not\in J_{1}$. Now by Lemma 4.5
there is a solution to $(\operatorname{ad}e)^{2}(f)=e$, where
$f=\sum_{i=1}^{n-1}a_{i}f_{i}$. Thus $f$ is in the type $A_{n-1}$ subalgebra
$\mathfrak{m}$ corresponding to the first $n-1$ simple roots and $e$ is a
standard representative in $\mathfrak{m}$. From Lemma 4.6 it follows that
$\mathcal{S}(e)$ has no $k_{i}\cong 2\pmod{4}$. Conversely, any other
$\mathcal{S}(e)$ with $s=t=0$ has a solution for $f$ by the same token.∎

######  Lemma 4.8.

Let $G\cong\mathrm{Spin}_{2n}$ with $n\not\equiv 2\pmod{4}$ and
$\mathcal{S}(e)=((n),(),())$ or $((),(l),(n))$ for any $\frac{n+1}{2}<l\leq
n$. Then $e$ is contained in an $\mathfrak{s}$-subalgebra of $\mathfrak{g}$,
and so Theorem 4.1 holds in these cases.

###### Proof.

In the first case, $e=\sum_{i=1}^{n-1}e_{i}$ is contained in a subalgebra of
type $A_{n-1}$. Now Lemma 4.6 yields an $\mathfrak{s}$-overalgebra of $e$,
since $n-1\not\equiv 1\pmod{4}$.

In the second case, Table 4 exhibits an $\mathfrak{s}$-overalgebra of $e$ with
structure constants as in (7). In each case, it is a routine calculation,
using the relations in Section 1 to check that $\langle
e,h,f\rangle\cong\mathfrak{s}$. ∎

$n$ | $h$ | $f_{0}$ | $l$ | $f$
---|---|---|---|---
$0\pmod{4}$ | $\sum\limits_{i=0}^{n/4-1}h_{4i+1}+h_{4i+2}$ | $\sum\limits_{i=0}^{n/4-1}f_{4i+1}+f_{4i+2}$ | even | $f_{0}$
|  |  | odd | $f_{0}+\sum\limits_{k=1}^{n-l}e_{\delta_{k}}$
$1\pmod{4}$ | $\sum\limits_{i=0}^{(n-5)/4}h_{4i+2}+h_{4i+3}$ | $\sum\limits_{i=0}^{(n-5)/4}f_{4i+2}+f_{4i+3}$ | even | $f_{0}+\sum\limits_{k=1}^{n-l}e_{\delta_{k}}$
|  |  | odd | $f_{0}$
$3\pmod{4}$ | $\sum\limits_{i=0}^{(n-3)/4}h_{4i+1}+h_{4i+2}$ | $\sum\limits_{i=0}^{(n-3)/4}f_{4i+1}+f_{4i+2}$ | even or $n$ | $f_{0}$
|  |  | otherwise |  $f_{0}+\sum\limits_{k=1}^{n-l}e_{\delta_{k}}$,

$\text{where}\qquad\delta_{k}=\sum\limits_{i=2l-n+k}^{n-2}\alpha_{i}+\sum\limits_{j=n-k+1}^{n}\alpha_{j},\quad
1\leq k\leq n-l.$

Table 4. $\mathfrak{s}$-overalgebras in $D_{n}$ when
$\mathcal{S}(e)=((),(l),(n))$

We can now deal with the general case.

######  Lemma 4.9.

Theorem 4.1 holds when $G\cong\mathrm{Spin}_{2n}$.

###### Proof.

Let $\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{i})_{i=1}^{s},(m_{i})_{i=1}^{s})$,
so that $e$ is contained in a commuting sum
$\mathfrak{m}=\mathfrak{m}_{1}+\dots+\mathfrak{m}_{r+s}$ of subalgebras of
type $A_{k_{1}-1},\ldots,A_{k_{r}-1}$, $D_{m_{1}},\ldots,D_{m_{s}}$. If all
$k_{i}\not\equiv 2\pmod{4}$ and all $m_{i}\not\equiv 2\pmod{4}$ then Lemmas
4.6 and 4.8 promise an $\mathfrak{s}$-overalgebra $\mathfrak{s}_{j}$ of the
projection of $e$ to each $\mathfrak{m}_{j}$. Then the diagonal subalgebra
$\mathfrak{s}\subseteq\mathfrak{s}_{1}\oplus\dots\oplus\mathfrak{s}_{r+s}$ is
an $\mathfrak{s}$-overalgebra of $e$.

On the other hand, suppose there is $k_{i}\equiv 2\pmod{4}$ or $m_{i}\equiv
2\pmod{4}$ and $e$ has an $\mathfrak{s}$-overalgebra. Consider the composition
$\varphi:G\to\mathrm{SO}_{2n}\to\mathrm{SL}_{2n}$, where the first map is a
central isogeny and the second is inclusion. This gives
$\mathrm{d}\varphi(\mathfrak{s})$ as an $\mathfrak{s}$-overalgebra of
$e^{\prime}:=\mathrm{d}\varphi(e)$. The Jordan blocks of $e^{\prime}$ can be
read off from the restriction of the natural $\mathrm{SL}_{2n}$ module $V$ to
$\Bbbk[e^{\prime}]$ in Table 2; we find $e^{\prime}$ has blocks of size
$k_{i}$ or $m_{i}$ on $V$. This contradicts Lemma 4.6. ∎

We complete this section with two results covering the type $B$ cases.

######  Lemma 4.10.

Suppose $G\cong\mathrm{Spin}_{2n+1}$ with $n\equiv 0$ or $3\pmod{4}$ and
suppose that $e$ is a regular nilpotent element of $\mathfrak{g}$. Then $e$ is
contained in an $\mathfrak{s}$-subalgebra of $\mathfrak{g}$ and Theorem 4.1
holds in this case.

###### Proof.

Since $e$ is regular, we have $\mathcal{S}(e)=((),(),(),n+1)$. We use the
embedding $\varphi:G=M^{\tau}\to M:=\mathrm{Spin}_{2n+2}$ where $G$ is the
stabiliser of the graph automorphism $\tau$ of $M$. Let $V$ be the natural
$(2n+2)$-dimensional module for $M$ and let
$e^{\prime}:=\mathrm{d}\varphi(e)$. As explained in Section 2.5,
$\mathcal{S}(e)=((),(n+1),(n+1))$. Since $n+1\equiv 1,0\pmod{4}$, Table 4
yields an $\mathfrak{s}$-overalgebra of $e^{\prime}$. Furthermore, in both
bases the relevant $f$ is simply $f_{0}$, which is easily seen to be stable
under the graph automorphism and hence contained in $\mathfrak{g}$. ∎

######  Lemma 4.11.

Theorem 4.1 holds for $G\cong\mathrm{Spin}_{2n+1}$.

###### Proof.

The proof is similar to that of Lemma 4.9, so we provide a sketch of the
details. Let
$\mathcal{S}(e)=((k_{i})_{i=1}^{r},(l_{j})_{j=1}^{s},(m_{k})_{k=1}^{s},m)$, so
that $e$ is contained in a commuting sum
$\mathfrak{m}=\mathfrak{m}_{1}+\dots+\mathfrak{m}_{r+s+1}$ of subalgebras of
type $A_{k_{1}-1},\ldots,A_{k_{r}-1}$, $D_{m_{1}},\ldots,D_{m_{s}}B_{m-1}$.

If all $k_{i}\not\equiv 2\pmod{4}$ and all $m_{i}\not\equiv 2\pmod{4}$ then
Lemmas 4.6 and 4.8 promise an $\mathfrak{s}$-overalgebra $\mathfrak{s}_{j}$ of
the projection of $e$ to each $\mathfrak{m}_{j}$. Similarly, Lemma 4.10
promises an $\mathfrak{s}$-overalgebra $\mathfrak{s}_{r+s+1}$ of the
projection of $e$ to $\mathfrak{m}_{r+s+1}$ when $m-1\not\equiv 1,2\pmod{4}$.
So when all three congruence conditions hold, the diagonal subalgebra
$\mathfrak{s}\subseteq\mathfrak{s}_{1}\oplus\dots\oplus\mathfrak{s}_{r+s}$ is
an $\mathfrak{s}$-overalgebra of $e$.

For the remaining cases consider the composition
$\varphi:G\to\mathrm{SO}_{2n+1}\to\mathrm{SL}_{2n+1}$, where the first map is
a central isogeny and the second is inclusion. Again
$\mathrm{d}\varphi(\mathfrak{s})$ is an $\mathfrak{s}$-overalgebra of
$e^{\prime}:=\mathrm{d}\varphi(e)$. This time we find $e^{\prime}$ has blocks
of size $k_{i}$, $m_{i}$, $m$ or $m-1$ on $V$. ∎

##  5\. Extending nilpotent elements to $\mathfrak{pgl}_{2}$-subalgebras

The group $G$ returns as a simple $\Bbbk$-algebra and $0\neq e\in\mathfrak{g}$
is nilpotent. We determine when $e$ extends to a $\mathfrak{pgl}_{2}$-triple
$(e,h,f)$, where $[e,f]=0$, $[h,e]=e$ and $[h,f]=f$. Note that if
$e^{\prime}\in\mathfrak{pgl}_{2}$ is nilpotent then
$e^{\prime}\in\mathcal{N}(\mathfrak{g})=[\mathfrak{pgl}_{2},\mathfrak{pgl}_{2}]$,
and by change of basis we may assume $e=e^{\prime}$.

######  Theorem 5.1.

The nonzero element $e$ extends to a $\mathfrak{pgl}_{2}$-triple $(e,h,f)$ if
and only if

  1. (i)

$G$ is not of type $A_{2}$ and there exists $h\in\mathfrak{g}$ such that
$[h,e]=e$ (see Theorem 3.1);

  2. (ii)

$G$ is of type $A_{2}$ and $e$ is not regular.

Again we postpone the calculations for exceptional $G$ to Section 7. If $e$
extends to a $\mathfrak{pgl}_{2}$-triple then there is $h$ with $[h,e]=e$, so
we may assume

$G$ is classical, $e$ is a standard representative from Table
LABEL:tab:stdreps and $(G,e)$ is listed in Theorem 3.1.

######  Lemma 5.2.

Theorem 5.1 holds when $G$ is of type $A_{n}$.

###### Proof.

The result is clear when $n=1$. Suppose $n=2$; then
$\mathfrak{g}\cong\mathfrak{sl}_{3}$. Suppose also that $e=e_{1}+e_{2}$ is
regular and $e$ extends to a $\mathfrak{pgl}_{2}$-triple. For ease of
notation, write $e_{3}=e_{\tilde{\alpha}}$ and $f_{3}=e_{-\tilde{\alpha}}$.
Then an easy calculation yields $C_{\mathfrak{g}}(e)=\langle e,e_{3}\rangle$,
so without loss of generality, $f=e_{3}$. We have assumed there is some
$h=\sum_{i}a_{i}e_{i}+\sum_{j}b_{j}h_{j}+\sum_{k}c_{k}f_{k}$ with $[h,e]=e$
and $[h,f]=f$. Now

$[h,e]=b_{2}e_{1}+b_{1}e_{2}+(a_{1}+a_{2})e_{3}+c_{3}(f_{1}+f_{2})+c_{1}h_{1}+c_{2}h_{2},$
$[h,f]=c_{3}(h_{1}+h_{2})+c_{2}e_{1}+c_{1}e_{2}+(b_{1}+b_{2})e_{3}.$

The equation $[h,e]=e$ forces $c_{i}=0$ for all $i$, $a_{1}=a_{2}$ and
$b_{1}=b_{2}=1$. But then $[h,f]=0$, a contradiction that establishes the
$n=2$ case.

Now suppose $n\geq 3$. When $[X(T):\mathbb{Z}\Phi]$ is coprime to $2$ we may
assume $G$ is adjoint; so $h=\sum_{i=1}^{n}h_{i}$ solves $[h,e]=e$. If $n$ is
odd we then define $f=e_{\tilde{\alpha}}$ and check $[e,f]=0$ and $[h,f]=f$.

When $n$ is even, there are two further cases. If $\alpha_{n}\not\in J$ then
$e$ is contained in a subalgebra $\mathfrak{l}$ of type $A_{n-1}$ with
$n-1\geq 3$ odd, in which case we have already exhibited a
$\mathfrak{pgl}_{2}$-triple for $e$. If $\alpha_{n}\in J$, choose
$f=e_{\alpha_{1}+\cdots+\alpha_{n-1}}+e_{\alpha_{2}+\cdots+\alpha_{n}}$. Then
$[h,f]=f$ and
$[e_{\alpha_{1}+\cdots+\alpha_{n-1}},e]=[e_{\alpha_{2}+\cdots+\alpha_{n}},e]=e_{\tilde{alpha}}$
so $[e,f]=0$ (noting that $\alpha_{1}\in J$).

Now suppose $[X(T):\mathbb{Z}\Phi]$ is divisible by $2$. Then $n\not\equiv
1\pmod{4}$ or $\mathcal{S}(e)=((k_{i})_{i=1}^{r})$ admits at least one odd
$k_{i}$. Since $\mathfrak{z}(\mathfrak{pgl}_{2})=0$, it will suffice to
construct a $\mathfrak{pgl}_{2}$-triple in $\mathfrak{sl}_{n+1}$.

We take $h$ from the proof of Lemma 3.6. In each case
$[h,e_{\delta}]=\text{ht}(\delta)e_{\delta}$. Then when $n$ is odd we may take
$f=e_{\tilde{\alpha}}$. When $n$ is even we define $f$ as follows and leave
the verification to the reader.

$f=\begin{cases}e_{\alpha_{1}+\cdots+\alpha_{n-1}}&\text{ if
}\alpha_{n}\not\in J,\\\
e_{\alpha_{1}+\cdots+\alpha_{n-1}}+e_{\alpha_{2}+\cdots+\alpha_{n}},&\text{ if
}\alpha_{n}\in J.\end{cases}\qed$

We may now assume until the end of the section:

$G$ has type $B$, $C$ or $D$.

######  Lemma 5.3.

Theorem 5.1 holds when $G$ is adjoint or $G\cong\mathrm{SO}_{2n}$.

###### Proof.

Let $h=\sum_{i}h_{i}$ and $f=e_{\tilde{\alpha}}$. The proof of Lemma 3.3 shows
$[h,e]=e$ and $[e,f]=0$ since $f\in\mathfrak{z}(\mathfrak{g}_{+})$. Finally,
since $\mathrm{ht}(\tilde{\alpha})$ is odd, (CB2) (or specifically (3)) show
that $[h,f]=f$. ∎

######  Lemma 5.4.

Theorem 5.1 holds when $G\cong\mathrm{Sp}_{2n}$.

###### Proof.

Since there is $h$ with $[h,e]=e$, we have
$e\in[\operatorname{Lie}(G),\operatorname{Lie}(G)]\cong\operatorname{Lie}(\mathrm{SO}_{2n})$
so we are done by Lemma 5.3. ∎

######  Lemma 5.5.

Theorem 5.1 holds when $G$ is of type $D$.

###### Proof.

It suffices to exhibit a $\mathfrak{pgl}_{2}$-triple for $e$ when
$G\cong\mathrm{Spin}_{2n}$, since $\mathfrak{pgl}_{2}$ is centre-free. In that
case, set $f=e_{\tilde{\alpha}}$ and $h$ as in the proof of Lemma 3.7. We
leave the verification to the reader. ∎

######  Lemma 5.6.

Theorem 5.1 holds when $G\cong\mathrm{Spin}_{2n+1}$.

###### Proof.

First assume $n\equiv 0,3\pmod{4}$. Choose $h$ as in the proof of Lemma 3.8.
Define $f=e_{\tilde{\alpha}}$ which satisfies $[h,f]=f$ thanks again to
$\tilde{\alpha}$ having an odd height. Furthermore, $f\in Z(\mathfrak{g}_{+})$
so $[e,f]=0$.

Now suppose $n\equiv 1,2\pmod{4}$ and there is a solution to $[h,e]=e$. As in
the proof of Lemma 3.8, we assume $e=y_{1}+y_{2}$ is a sum of standard
nilpotent elements contained in a subsystem subalgebra
$\mathfrak{m}_{1}+\mathfrak{m}_{2}$ with $y_{1}\in\mathfrak{m}_{1}$ of type
$D_{x}$, $x$ odd and $y_{2}\in\mathfrak{m}_{2}$ of type $B_{n-x}$.
Furthermore, Lemma 5.5 and the previous paragraph yield
$\mathfrak{pgl}_{2}$-overalgebras of $y_{i}\in\mathfrak{m}_{i}$ unless
$(n,x)\equiv(1,3)$ or $(2,1)\pmod{4}$. In the remaining two cases an $h$ is
provided in the proof of Lemma 3.8 that satisfies $[h,y_{i}]=y_{i}$. Thanks to
the choice of subsystem subalgebra and $y_{i}$ being in standard form,
$e=y_{1}+y_{2}\in\mathfrak{g}_{\geq 1}$. We break off into two cases. If
$x\neq 1$ then set $f=e_{\tilde{\alpha}}$, so $[y_{1}+y_{2},f]=0$. It follows
from Equation (6) that $[h,f]=f$ showing we have found a
$\mathfrak{pgl}_{2}$-subalgebra containing $e$.

Finally, suppose $x=1$, so $n\equiv 2\pmod{4}$. Then $e$ is contained entirely
within the subalgebra $\mathfrak{m}_{2}$ of type $B_{n-1}$, where $n-1\equiv
1\pmod{4}$. We have already proven that $e$ has a
$\mathfrak{pgl}_{2}$-overalgebra in $\mathfrak{m}_{2}$, completing the proof.
∎

##  6\. Extending nilpotent elements to $\mathfrak{sl}_{2}$-subalgebras

The simple $k$-group $G$ returns to arbitrary type. In this section we find
when $e\in\mathcal{N}$ can be extended to an $\mathfrak{sl}_{2}$-triple
$(e,h,f)$ with

$[e,f]=h,\ \ [h,e]=[h,f]=0.$

In other words, we find when $e$ sits inside a $3$-dimensional Heisenberg
algebra.

######  Theorem 6.1.

The nonzero nilpotent element $e$ extends to an $\mathfrak{sl}_{2}$-triple if
and only if $G$ is not isomorphic to $\mathrm{PGL}_{2}$.

###### Proof.

The case that $G$ has rank $1$ is clear, since $\mathfrak{sl}_{2}$ is not a
subalgebra of $\mathfrak{pgl}_{2}$. As usual, exceptional types are postponed
to Section 7 and we are left to deal with classical types of rank at least
$2$.

For each non-zero standard nilpotent representative $e\in\mathfrak{g}$ we
exhibit an $\mathfrak{sl}_{2}$ overalgebra in $\mathfrak{g}_{+}$. Since
$\mathfrak{g}_{+}\cap\mathfrak{z}(\mathfrak{g})=0$, the map
$\mathfrak{g}\to\mathfrak{g}/\mathfrak{z}(\mathfrak{g})$ induces a bijection
on the respective nilpotent cones. Thus we may assume $G$ is simply connected
and $e$ is a standard representative supported on roots $J$. Note that since
$e$ is non-zero, $\alpha_{1}\in J$. We proceed by exhibiting
$f\in\mathfrak{g}_{+}$ such that $\langle e,f\rangle\cong\mathfrak{sl}_{2}$.

For types $A$ and $C$, define

$f=e_{\tilde{\alpha}-\alpha_{1}}.$

Since $\alpha_{1}\in J$, it follows that $[e,f]=e_{\tilde{\alpha}}$. Since
$e_{\tilde{\alpha}}\in\mathfrak{z}(\mathfrak{g}_{+})$, we are done.

For types $B$ and $D$ we differentiate between two cases and define

$f=\begin{cases}e_{\tilde{\alpha}-\alpha_{2}}&\text{ if }\alpha_{2}\in J,\\\
e_{\tilde{\alpha}-\alpha_{1}-\alpha_{2}},&\text{ if }\alpha_{2}\not\in
J.\end{cases}$

Then $[e,f]=e_{\tilde{\alpha}}$, $e_{\tilde{\alpha}-\alpha_{1}}$,
respectively. It is straightforward to check that $e$ and $f$ are in the
centraliser of $[e,f]$ in both cases. ∎

##  7\. Main theorems for exceptional groups

In this last section, we let $G$ be a simple $k$-group of exceptional type. We
describe the calculations which establish Theorems 3.1, 4.1, 5.1 and 6.1 for
$G$, completing their proofs. The calculations were done in Magma, and the
code can be found at https://github.com/davistem/nilpotent_orbits_magma.

###  7.1. Nilpotent orbit representatives

Any nilpotent element $e\in\mathfrak{g}$ is conjugate to an element in
$\mathfrak{g}_{+}$. Since $\mathfrak{g}_{+}$ has the positive root vectors as
a basis, each orbit $\mathrm{O}\subseteq\mathfrak{g}$ is represented by a
linear combination of positive root vectors. It turns out that one can always
give a representative which is a sum of at most $n+1$ root vectors, where $n$
is the rank of $G$. For a full description of how these are obtained, labelled
and verified, see [KST22] or LS04.

In practice, we have a function which given an exceptional type and isogeny
produces a Lie algebra $\mathfrak{g}_{\mathbb{F}_{2}}$ over $\mathbb{F}_{2}$
such that
$\mathfrak{g}_{\mathbb{F}_{2}}\otimes_{\mathbb{F}_{2}}\Bbbk=\mathfrak{g}$,
together with representatives of nilpotent orbits and their labels. The
algebra $\mathfrak{g}_{\mathbb{F}_{2}}$ comes with a canonical Chevalley basis
whose associated structure constants are exactly as described in Section 2.1,
and the representatives are as sums of of these basis elements. For example:

⬇

> Read("exceptional_code_functions.m");

> g,orbs,labels:=Liealgreps("G2","Ad");

> E,F,H:=ChevalleyBasis(g);

> e:=orbs[1]; e;

(0 0 0 0 0 0 0 0 1 1 0 0 0 0)

> labels[1];

"G_2"

Even when $\mathfrak{g}$ is of type $E_{8}$ there are only 75 orbits to deal
with and the following calculations are all handled in seconds.

###  7.2. Solving equations in $\mathfrak{g}$

For $e\in\mathfrak{g}$ nilpotent, there is a solution to $[h,e]=e$ if and only
if $e\in\operatorname{im}\operatorname{ad}(e)$. Since
$e\in\mathfrak{g}_{\mathbb{F}_{2}}$, the solution set
$S:=\\{h\in\mathfrak{g}\mid[h,e]=e\\}$ is stable under
$\mathrm{Gal}(k/\mathbb{F}_{2})$; thus $[h,e]=e$ has a solution in
$\mathfrak{g}$ if and only if it has one in $\mathfrak{g}_{\mathbb{F}_{2}}$.
Letting $E$ be the matrix of the linear map $\operatorname{ad}(e)$ relative to
a given basis of $\mathfrak{g}$, we use Magma’s linear algebra capabilities to
check when $e\in\operatorname{im}E$.

⬇

> g,orbs,labels:=Liealgreps("E7","SC");

> Ad:=AdjointRepresentation(g);

> e:=orbs[1];

> M:=Matrix(Ad(e));

> e in Image(M);

false

Repeating this process over all representatives of nilpotent orbits, we
establish Theorem 3.1.

Theorem 4.1 is similar. We have $e$ contained in an $\mathfrak{s}$-subalgebra
only if $e\in\operatorname{im}E^{2}$. We exclude those orbits which fail to
have a solution. For those that succeed, Magma provides a solution for
$E^{2}(f)=e$ with $f$ expressed in terms of the given basis. In each case, it
turned out that the $f$ supplied satisfied $[h,f]=f$ and $[h,e]=e$, where
$h:=[e,f]$. For example,

⬇

> g,orbs,labels:=Liealgreps("E7","Ad");

> E,F,H:=ChevalleyBasis(g);

> V:=VectorSpace(g);

> Ad:=AdjointRepresentation(g);

> e:=orbs[5];

> M:=Matrix(Ad(e));

> e in Image(M^2);

true

> v:=Solution(M^2,e);

> f:=g!(V!v);

> h:=e*f;

> h*e eq e;

true

> h*f eq f;

true

If $e\in\mathfrak{pgl}_{2}$ then we must have $[h,e]=e$. For each such
representative, we are always able to find $f\in\mathfrak{g}_{+}$ with
$[h,f]=f$ and $[e,f]=0$. (Magma looks to use a single positive root vector for
$f$ and is always successful.) This deals with Theorem 5.1.

Lastly, for Theorem 6.1, we ask Magma to look for a root vector $f$ such that
$[e,f]\neq 0$ and $[e,f]$ commutes with $e$ and $f$. Happily one is always
found.

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