# New Einstein metrics on the Lie group $\operatorname{SO}(n)$ which are not
naturally reductive

Andreas Arvanitoyeorgos, Yusuke Sakane, and Marina Statha  University of
Patras, Department of Mathematics, GR-26500 Patras, Greece
<EMAIL_ADDRESS>Osaka University, Department of Pure and Applied
Mathematics, Graduate School of Information Science and Technology, Toyonaka,
Osaka 560-0043, Japan<EMAIL_ADDRESS>University of Patras,
Department of Mathematics, GR-26500 Patras, Greece
<EMAIL_ADDRESS>
###### Abstract.

We obtain new invariant Einstein metrics on the compact Lie groups
$\operatorname{SO}(n)$ ($n\geq 7$) which are not naturally reductive. This is
achieved by imposing certain symmetry assumptions in the set of all left-
invariant metrics on $\operatorname{SO}(n)$ and by computing the Ricci tensor
for such metrics. The Einstein metrics are obtained as solutions of systems
polynomial equations, which we manipulate by symbolic computations using
Gröbner bases.

2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C30,
22C05, 22E60, 13P05, 13P10, 13P15

Keywords: Homogeneous space; Einstein metric; isotropy representation; compact
simple Lie group; naturally reductive metric; generalized Wallach space;
algebraic system of equations; Gröbner basis

##  1\. Introduction

A Riemannian manifold $(M,g)$ is called Einstein if it has constant Ricci
curvature, i.e. $\operatorname{Ric}_{g}=\lambda\cdot g$ for some
$\lambda\in\mathbb{R}$. For results on Einstein manifolds before 1987 we refer
to the book by A. Besse [Be]. The two articles [Wa1], [Wa2] of M. Wang contain
results up to 1999 and 2013 respectively. General existence results are
difficult to obtain and some methods are described in [Bö], [BöWaZi] and
[WaZi]. For homogeneous spaces $G/K$ the problem is to find and classify all
$G$-invariant Einstein metrics. The problem is even more difficult for the
case of a Lie group, where we need to find (or prove existence) of left-
invariant Einstein metrics. Even for the compact Lie groups
$\operatorname{SU}(3)$ and $\operatorname{SU}(2)\times\operatorname{SU}(2)$
the number of left-invariant Einstein metrics is still unknown.

In the present paper we investigate left-invariant Einstein metrics on the
compact Lie group $\operatorname{SO}(n)$. It is known that a compact and
semisimple Lie group equipped with a bi-invariant metric is Einstein. In the
work [DAZi], J.E. D’Atri and W. Ziller found a large number of left-invariant
Einstein metrics, which are naturally reductive, on the compact Lie groups
$\operatorname{SU}(n),\operatorname{SO}(n)$ and $\operatorname{Sp}(n)$. In the
same article they posed the question whether there exist left-invariant
Einstein metrics on compact Lie groups which are not naturally reductive.

Some contributions to this problem are the following: In [Mo] K. Mori obtained
non naturally reductive Einstein metrics on the Lie group
$\operatorname{SU}(n)$ for $n\geq 6$, and in [ArMoSa] the first two authors
and K. Mori proved existence of non naturally reductive Einstein metrics on
the compact Lie groups $\operatorname{SO}(n)$ $(n\geq 11)$,
$\operatorname{Sp}(n)$ $(n\geq 3)$, $E_{6},E_{7}$ and $E_{8}$. In [ChLi] Z.
Chen and K. Liang found three naturally reductive and one non naturally
reductive Einstein metric on the compact Lie group $F_{4}$. Also, in [ArSaSt2]
the authors obtained new left-invariant Einstein metrics on the symplectic
group $\operatorname{Sp}(n)\ (n\geq 3)$, and in [ChSa] I. Chrysikos and the
second author obtained non naturally reductive Einstein metrics on exceptional
Lie groups. We also mention the works [GiLuPo], [Po] by G.W. Gibbons, H. Lü
and C. N. Pope where they discussed left-invariant Einstein metrics on the Lie
groups $\operatorname{SO}(n)$, $G_{2}$ and $\operatorname{SU}(3)$ which are
however naturally reductive, as well as [Mu] by A. H. Mujtaba, who obtained
certain classes of left-invariant metrics on $\operatorname{SU}(n)$, which
were previously found in [Je2] and [DAZi].

The aim of the present work is to obtain left-invariant Einstein metrics on
the compact Lie groups $\operatorname{SO}(n)$ $(n\geq 7)$ which are not
naturally reductive. The Einstein metrics obtained here are different from the
ones obtained in [ArMoSa]. The idea behind our approach is to consider an
appropriate subgroup $K$ and a corresponding homogeneous space $G/K$ whose
isotropy representation decomposes into $\operatorname{Ad}(K)$-irreducible and
non equivalent summands. Then the tangent space $\mathfrak{g}$ of $G$
decomposes, via the submersion $G\to G/K$ with fiber $K$, into a direct sum of
non equivalent $\operatorname{Ad}(K)$-modules. By taking into account the
diffeomorphism $G/\\{e\\}\cong(G\times K)/\mbox{diag}(K)$ we consider left-
invariant metrics on $G$ which are determined by diagonal
$\operatorname{Ad}(K)$-invariant scalar products on $\mathfrak{g}$, which in
turn enables us to use well known formulas for the Ricci curvature (e.g. [Be,
Corollary 7.38, p. 184], [PaSa, Lemma 1.1, p. 52]).

More precisely, for the group $\operatorname{SO}(n)$ we write
$n=k_{1}+k_{2}+k_{3}$, ($k_{1},k_{2},k_{3}$ positive integers), and we
consider the closed subgroup
$K=\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})$.
This determines the homogeneous space
$G/K=\operatorname{SO}(k_{1}+k_{2}+k_{3})/(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$,
which is an example of a generalized Wallach space according to [NiRoSla]. For
$k_{1}\geq k_{2}\geq k_{3}\geq 2$ with $k_{i}\neq k_{j}$ the isotropy
representation
$\mathfrak{m}=\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}$
of $G/K$ does not contain equivalent summands, thus we consider left-invariant
metrics determined by
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
inner products of the form

$\begin{array}[]{lll}\langle\ ,\
\rangle&=&x_{1}\,(-B)|_{\mathfrak{so}(k_{1})}+x_{2}\,(-B)|_{\mathfrak{so}(k_{2})}+x_{3}\,(-B)|_{\mathfrak{so}(k_{3})}\\\
&&+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}}\end{array},\quad
x_{i},x_{ij}>0.$

If $k_{3}=1$, $k_{1}\geq k_{2}\geq 2$ we omit the variable $x_{3}$, and if
$k_{2}=k_{3}=1$, $k_{1}\geq 3$ we omit the variables $x_{2}$ and $x_{3}$. In
the last case the submodules $\mathfrak{m}_{12}$ and $\mathfrak{m}_{13}$ are
equivalent and we treat this separately in Section 7.

By using the main theorem of D’Atri and Ziller ([DAZi]) we obtain conditions
on the positive variables $x_{1},x_{2},x_{3},x_{12},x_{13}$ and $x_{23}$ so
that the above metric is naturally reductive. For all possible partitions of
$n=k_{1}+k_{2}+k_{3}$ we write the components of the Ricci tensor and then use
methods of symbolic computation to prove existence of positive solutions to
systems of algebraic equations obtained by the Einstein equation. For the case
of the Lie groups $\operatorname{SO}(5)$ and $\operatorname{SO}(6)$ our method
gives only naturally reductive metrics.

The main result is the following:

######  Theorem 1.1.

The compact simple Lie groups $\operatorname{SO}(n)$ $(n\geq 7)$ admit left-
invariant Einstein metrics which are not naturally reductive.

The paper is organized as follows: In Section 2 we recall a formula for the
Ricci tensor of homogeneous spaces given in [PaSa]. In Section 3 we describe
the left-invariant metrics which will be considered in this work, and in
Section 4 we give conditions under which such left-invariant metrics on
$\operatorname{SO}(n)$ are naturally reductive with respect to
$\operatorname{SO}(n)\times L$, for some closed subgroup $L$ of
$\operatorname{SO}(n)$. In Section 5 we obtain explicit formulas for the Ricci
tensor of left-invariant metrics. In Section 6 we investigate the solutions of
the Einstein equation by using Gröbner bases. Then Theorem 1.1 follows from
Propositions 6.1, 6.2 and 6.3. Finally, in Section 7 we prove that the compact
Lie groups $\operatorname{SO}(n)$ admit only naturally reductive
$\operatorname{Ad}(\operatorname{SO}(n-2))$-invariant Einstein metrics (here
$k_{1}=n-2,k_{2}=k_{3}=1$). This case is of special interest, because the
decomposition
$\mathfrak{so}(n)=\mathfrak{so}(k_{1})\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}$
of the tangent space of $\operatorname{SO}(n)$ contains equivalent
$\operatorname{Ad}(\operatorname{SO}(n-2))$-modules. Hence, we need to confirm
that for the $\operatorname{Ad}(\operatorname{SO}(n-2))$-invariant metrics
under consideration the Ricci tensor is diagonal.

Acknowledgements. The work was supported by Grant $\\#E.037$ from the Research
Committee of the University of Patras (Programme K. Karatheodori) and JSPS
KAKENHI Grant Number 25400071. It was completed while the first author was on
sabbatical leave at Tufts University, University of Athens and Osaka
University during 2014-15.

##  2\. The Ricci tensor for reductive homogeneous spaces

In this section we recall an expression for the Ricci tensor for an
$G$-invariant Riemannian metric on a reductive homogeneous space whose
isotropy representation is decomposed into a sum of non equivalent irreducible
summands.

Let $G$ be a compact semisimple Lie group, $K$ a connected closed subgroup of
$G$ and let $\mathfrak{g}$ and $\mathfrak{k}$ be the corresponding Lie
algebras. The Killing form $B$ of $\mathfrak{g}$ is negative definite, so we
can define an $\mbox{Ad}(G)$-invariant inner product $-B$ on $\mathfrak{g}$.
Let $\mathfrak{g}$ = $\mathfrak{k}\oplus\mathfrak{m}$ be a reductive
decomposition of $\mathfrak{g}$ with respect to $-B$ so that
$\left[\,\mathfrak{k},\,\mathfrak{m}\,\right]\subset\mathfrak{m}$ and
$\mathfrak{m}\cong T_{o}(G/K)$. We assume that ${\mathfrak{m}}$ admits a
decomposition into mutually non equivalent irreducible $\mbox{Ad}(K)$-modules
as follows:

${\mathfrak{m}}={\mathfrak{m}}_{1}\oplus\cdots\oplus{\mathfrak{m}}_{q}.$ (1)

Then any $G$-invariant metric on $G/K$ can be expressed as

$\displaystyle\langle\ ,\
\rangle=x_{1}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{1}}+\cdots+x_{q}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{q}},$
(2)

for positive real numbers $(x_{1},\dots,x_{q})\in\mathbb{R}^{q}_{+}$. Note
that $G$-invariant symmetric covariant 2-tensors on $G/K$ are of the same form
as the Riemannian metrics (although they are not necessarilly positive
definite). In particular, the Ricci tensor $r$ of a $G$-invariant Riemannian
metric on $G/K$ is of the same form as (2), that is

$r=y_{1}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{1}}+\cdots+y_{q}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{q}},$

for some real numbers $y_{1},\ldots,y_{q}$.

Let $\\{e_{\alpha}\\}$ be a $(-B)$-orthonormal basis adapted to the
decomposition of $\mathfrak{m}$, i.e. $e_{\alpha}\in{\mathfrak{m}}_{i}$ for
some $i$, and $\alpha<\beta$ if $i<j$. We put
${A^{\gamma}_{\alpha\beta}}=-B\left(\left[e_{\alpha},e_{\beta}\right],e_{\gamma}\right)$
so that
$\left[e_{\alpha},e_{\beta}\right]=\displaystyle{\sum_{\gamma}A^{\gamma}_{\alpha\beta}e_{\gamma}}$
and set $\displaystyle{k\brack{ij}}=\sum(A^{\gamma}_{\alpha\beta})^{2}$, where
the sum is taken over all indices $\alpha,\beta,\gamma$ with
$e_{\alpha}\in{\mathfrak{m}}_{i},\ e_{\beta}\in{\mathfrak{m}}_{j},\
e_{\gamma}\in{\mathfrak{m}}_{k}$ (cf. [WaZi]). Then the positive numbers
$\displaystyle{k\brack{ij}}$ are independent of the $B$-orthonormal bases
chosen for ${\mathfrak{m}}_{i},{\mathfrak{m}}_{j},{\mathfrak{m}}_{k}$, and
$\displaystyle{k\brack{ij}}\ =\ \displaystyle{k\brack{ji}}\ =\
\displaystyle{j\brack{ki}}.$

Let $d_{k}=\dim{\mathfrak{m}}_{k}$. Then we have the following:

######  Lemma 2.1.

([PaSa]) The components ${r}_{1},\dots,{r}_{q}$ of the Ricci tensor ${r}$ of
the metric $\langle\,\,\,,\,\,\,\rangle$ of the form (2) on $G/K$ are given by

${r}_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{j,i}\frac{x_{k}}{x_{j}x_{i}}{k\brack{ji}}-\frac{1}{2d_{k}}\sum_{j,i}\frac{x_{j}}{x_{k}x_{i}}{j\brack{ki}}\quad(k=1,\
\dots,q),$ (3)

where the sum is taken over $i,j=1,\dots,q$.

Since by assumption the submodules $\mathfrak{m}_{i},\mathfrak{m}_{j}$ in the
decomposition (1) are mutually non equivalent for any $i\neq j$, it is
$r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$ whenever $i\neq j$. If
$\mathfrak{m}_{i}\cong\mathfrak{m}_{j}$ for some $i\neq j$ then we need to
check whether $r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$. This is not an easy
task in general. Once the condition $r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$
is confirmed we can use Lemma 2.1. Then $G$-invariant Einstein metrics on
$M=G/K$ are exactly the positive real solutions
$g=(x_{1},\ldots,x_{q})\in\mathbb{R}^{q}_{+}$ of the polynomial system
$\\{r_{1}=\lambda,\ r_{2}=\lambda,\ \ldots,\ r_{q}=\lambda\\}$, where
$\lambda\in\mathbb{R}_{+}$ is the Einstein constant.

##  3\. A class of left-invariant metrics on
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$

We will describe a decomposition of the tangent space of the Lie group
$\operatorname{SO}(n)$ which will be convenient for our study. We consider the
closed subgroup
$K=\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})$
of $G=\operatorname{SO}(k_{1}+k_{2}+k_{3})$ ($k_{1}\geq k_{2}\geq k_{3}\geq
2$), where the embedding of $K$ in $G$ is diagonal, and the fibration

$\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})\to\operatorname{SO}(k_{1}+k_{2}+k_{3})\to\operatorname{SO}(k_{1}+k_{2}+k_{3})/(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})).$

The base space of the above fibration is an example of a generalized Wallach
space (cf. [NiRoSla]). Then the tangent space
$\mathfrak{so}(k_{1}+k_{2}+k_{3})$ of the orthogonal group
$G=\operatorname{SO}(k_{1}+k_{2}+k_{3})$ can be written as a direct sum of two
$\operatorname{Ad}(K)$-invariant modules, the horizontal space
$\mathfrak{m}\cong T_{o}(G/K)$ and the vertical space
$\mathfrak{so}(k_{1})\oplus\mathfrak{so}(k_{2})\oplus\mathfrak{so}(k_{3})$,
i.e.

$\mathfrak{so}(k_{1}+k_{2}+k_{3})=\mathfrak{so}(k_{1})\oplus\mathfrak{so}(k_{2})\oplus\mathfrak{so}(k_{3})\oplus\mathfrak{m}.$
(4)

The tangent space $\mathfrak{m}$ of $G/K$ is given by $\mathfrak{k}^{\perp}$
in $\mathfrak{g}=\mathfrak{so}(k_{1}+k_{2}+k_{3})$ with respect to $-B$. If we
denote by $M(p,q)$ the set of all $p\times q$ matrices, then we see that
$\mathfrak{m}$ is given by

$\mathfrak{m}=\left\\{\begin{pmatrix}0&{A}_{12}&{A}_{13}\\\
-{}^{t}\\!{A}_{12}&0&{A}_{23}\\\
-{}^{t}\\!{A}_{13}&-{}^{t}\\!{A}_{23}&0\end{pmatrix}\ \Big{|}\ {A}_{12}\in
M(k_{1},k_{2}),{A}_{13}\in M(k_{1},k_{3}),{A}_{23}\in M(k_{2},k_{3})\right\\}$
(5)

and we have

$\mathfrak{m}_{12}=\begin{pmatrix}0&{A}_{12}&0\\\ -{}^{t}\\!{A}_{12}&0&0\\\
0&0&0\end{pmatrix},\quad\mathfrak{m}_{13}=\begin{pmatrix}0&0&{A}_{13}\\\
0&0&0\\\
-{}^{t}\\!{A}_{13}&0&0\end{pmatrix},\quad\mathfrak{m}_{23}=\begin{pmatrix}0&0&0\\\
0&0&{A}_{23}\\\ 0&-{}^{t}\\!{A}_{23}&0\end{pmatrix}.$ (6)

Note that the action of $\operatorname{Ad}(k)$ ($k\in K$) on $\mathfrak{m}$ is
given by

$\operatorname{Ad}(k)\begin{pmatrix}0&{A}_{12}&{A}_{13}\\\
-{}^{t}\\!{A}_{12}&0&{A}_{23}\\\
-{}^{t}\\!{A}_{13}&-{}^{t}\\!{A}_{23}&0\end{pmatrix}=\begin{pmatrix}0&{}^{t}h_{1}{A}_{12}h_{2}&{}^{t}h_{1}{A}_{13}h_{3}\\\
-{}^{t}h_{2}{}^{t}\\!{A}_{12}h_{1}&0&{}^{t}h_{2}{A}_{23}h_{3}\\\
-{}^{t}h_{3}{}^{t}\\!{A}_{13}h_{1}&-{}^{t}h_{3}{}^{t}\\!{A}_{23}h_{2}&0\end{pmatrix},$
(7)

where $\begin{pmatrix}h_{1}&0&0\\\ 0&h_{2}&0\\\ 0&0&h_{3}\end{pmatrix}\in K$.
The subspaces $\mathfrak{m}_{12}$, $\mathfrak{m}_{13}$ and $\mathfrak{m}_{23}$
are irreducible $\operatorname{Ad}(K)$-submodules whose dimensions are
$\dim\mathfrak{m}_{12}=k_{1}k_{2}$, $\dim\mathfrak{m}_{13}=k_{1}k_{3}$ and
$\dim\mathfrak{m}_{23}=k_{2}k_{3}$. They are given as $(-B)$-orthogonal
complements of $\mathfrak{so}(k_{i})\oplus\mathfrak{so}(k_{j})$ in
$\mathfrak{so}(k_{i}+k_{j})$ ($1\leq i<j\leq 3$), respectively.

Note that the irreducible submodules $\mathfrak{m}_{ij}$ are mutually non
equivalent, so any $G$-invariant metric on the base space $G/K$ is determined
by an $\operatorname{Ad}(K)$-invariant scalar product
$x_{12}(-B)|_{\mathfrak{m}_{12}}$
$+x_{13}(-B)|_{\mathfrak{m}_{13}}+x_{23}(-B)|_{\mathfrak{m}_{23}}.$ We also
set
$\mathfrak{m}_{1}=\mathfrak{so}(k_{1}),\mathfrak{m}_{2}=\mathfrak{so}(k_{2})$
and $\mathfrak{m}_{3}=\mathfrak{so}(k_{3})$. Therefore, decomposition (4) of
the tangent space of the orthogonal group
$G=\operatorname{SO}(k_{1}+k_{2}+k_{3})$ takes the form

$\mathfrak{so}(k_{1}+k_{2}+k_{3})=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}.$
(8)

Let $E_{ab}$ denotes the $n\times n$ matrix with $1$ at the $(ab)$-entry and
$0$ elsewhere. Then the set $\mathcal{B}=\\{e_{ab}=E_{ab}-E_{ba}:1\leq a<b\leq
n\\}$ constitutes a $(-B)$-orthonormal basis of $\mathfrak{so}(n)$. Note that
$e_{ba}=-e_{ab}$, thus we have the following:

######  Lemma 3.1.

If all four indices are distinct, then the Lie brakets in $\mathcal{B}$ are
zero. Otherwise, $[e_{ab},e_{bc}]=e_{ac}$, where $a,b,c$ are distinct.

By using Lemma 3.1 we obtain:

######  Lemma 3.2.

The submodules in the decomposition (8) satisfy the following bracket
relations:

$[\mathfrak{m}_{1},\mathfrak{m}_{1}]=\mathfrak{m}_{1},$ | $[\mathfrak{m}_{2},\mathfrak{m}_{2}]=\mathfrak{m}_{2},$ | $[\mathfrak{m}_{3},\mathfrak{m}_{3}]=\mathfrak{m}_{3},$
---|---|---
$[\mathfrak{m}_{1},\mathfrak{m}_{12}]=\mathfrak{m}_{12},$ | $[\mathfrak{m}_{1},\mathfrak{m}_{13}]=\mathfrak{m}_{13},$ | $[\mathfrak{m}_{2},\mathfrak{m}_{12}]=\mathfrak{m}_{12},$
$[\mathfrak{m}_{2},\mathfrak{m}_{23}]=\mathfrak{m}_{23},$ | $[\mathfrak{m}_{3},\mathfrak{m}_{13}]=\mathfrak{m}_{13},$ | $[\mathfrak{m}_{3},\mathfrak{m}_{23}]=\mathfrak{m}_{23},$
$[\mathfrak{m}_{12},\mathfrak{m}_{23}]=\mathfrak{m}_{13},$ | $[\mathfrak{m}_{13},\mathfrak{m}_{23}]=\mathfrak{m}_{12},$ | $[\mathfrak{m}_{12},\mathfrak{m}_{13}]=\mathfrak{m}_{23}$
$[\mathfrak{m}_{12},\mathfrak{m}_{12}]=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2},$ | $[\mathfrak{m}_{13},\mathfrak{m}_{13}]=\mathfrak{m}_{1}\oplus\mathfrak{m}_{3},$ | $[\mathfrak{m}_{23},\mathfrak{m}_{23}]=\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}.$

Therefore, we see that the only non zero symbols (up to permutation of
indices) are

${1\brack{11}},{2\brack{22}},{3\brack{33}},{(12)\brack{1(12)}},{(13)\brack{1(13)}},{(12)\brack{2(12)}},{(23)\brack{2(23)}},{(13)\brack{3(13)}},{(23)\brack{3(23)}},{(13)\brack{(12)(23)}},$
(9)

where $\displaystyle{{i\brack{ii}}}$ is non zero only for $k_{i}\geq 3$
($i=1,2,3$).

Now we take into account the diffeomorphism

$G/\\{e\\}\cong(G\times\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))/{\rm
diag}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$

and consider left-invariant metrics on $G$ which are determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
scalar products on $\mathfrak{so}(k_{1}+k_{2}+k_{3})$ given by

$\begin{array}[]{lll}\langle\ ,\
\rangle&=&x_{1}\,(-B)|_{\mathfrak{so}(k_{1})}+x_{2}\,(-B)|_{\mathfrak{so}(k_{2})}+x_{3}\,(-B)|_{\mathfrak{so}(k_{3})}\\\
&&+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}}\end{array}.$
(10)

For $k_{3}=1$ we also consider left-invariant metrics on $G$ which are
determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2}))$-invariant
scalar products on $\mathfrak{so}(k_{1}+k_{2}+1)$ of the form

$\begin{array}[]{l}\langle\ ,\
\rangle=x_{1}\,(-B)|_{\mathfrak{so}(k_{1})}+x_{2}\,(-B)|_{\mathfrak{so}(k_{2})}+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}}.\end{array}$
(11)

Finally, for $k_{1}=n-2$ and $k_{2}=k_{3}=1$ we consider left-invariant
metrics on $G$ which are determined by the
$\operatorname{Ad}(\operatorname{SO}(n-2))$-invariant scalar products on
$\mathfrak{so}(n)$ of the form

$\begin{array}[]{l}\langle\ ,\
\rangle=x_{1}\,(-B)|_{\mathfrak{so}(n-2)}+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}}.\end{array}$
(12)

For the scalar products (11) the only non zero triplets are

${1\brack{11}},{2\brack{22}},{(12)\brack{1(12)}},{(13)\brack{1(13)}},{(12)\brack{2(12)}},{(23)\brack{2(23)}},{(13)\brack{(12)(23)}},$

and for the scalar products (12) the only non zero triplets are

${1\brack{11}},{(12)\brack{1(12)}},{(13)\brack{1(13)}},{(13)\brack{(12)(23)}}.$

##  4\. Naturally reductive metrics on the compact Lie groups
$\operatorname{SO}(n)$

A Riemannian homogeneous space $(M=G/K,g)$ with reductive complement
$\mathfrak{m}$ of $\mathfrak{k}$ in $\mathfrak{g}$ is called naturally
reductive if

$\langle[X,Y]_{\mathfrak{m}},Z\rangle+\langle
Y,[X,Z]_{\mathfrak{m}}\rangle=0\quad\mbox{for all}\ X,Y,Z\in\mathfrak{m}.$

Here $\langle\ ,\ \rangle$ denotes the inner product on $\mathfrak{m}$ induced
from the Riemannian metric $g$. Classical examples of naturally reductive
homogeneous spaces include irreducible symmetric spaces, isotropy irreducible
homogeneous manifolds, and Lie groups with bi-invariant metrics. In general it
is not always easy to decide if a given homogeneous Riemannian manifold is
naturally reductive, since one has to consider all possible transitive actions
of subgroups $G$ of the isometry group of $(M,g)$.

In [DAZi] D’Atri and Ziller had investigated naturally reductive metrics among
left-invariant metrics on compact Lie groups and gave a complete
classification in the case of simple Lie groups. Let $G$ be a compact,
connected semisimple Lie group, $L$ a closed subgroup of $G$ and let
$\mathfrak{g}$ be the Lie algebra of $G$ and $\mathfrak{l}$ the subalgebra
corresponding to $L$. We denote by $Q$ the negative of the Killing form of
$\mathfrak{g}$. Then $Q$ is an $\operatorname{Ad}(G)$-invariant inner product
on $\mathfrak{g}$.

Let $\mathfrak{m}$ be an orthogonal complement of $\mathfrak{l}$ with respect
to $Q$. Then we have

$\mathfrak{g}=\mathfrak{l}\oplus\mathfrak{m},\quad\operatorname{Ad}(L)\mathfrak{m}\subset\mathfrak{m}.$

Let
$\mathfrak{l}=\mathfrak{l}_{0}\oplus\mathfrak{l}_{1}\oplus\cdots\oplus{l}_{p}$
be a decomposition of $\mathfrak{l}$ into ideals, where $\mathfrak{l}_{0}$ is
the center of $\mathfrak{l}$ and $\mathfrak{l}_{i}$ $(i=1,\dots,p)$ are simple
ideals of $\mathfrak{l}$. Let $A_{0}|_{\mathfrak{l}_{0}}$ be an arbitrary
metric on $\mathfrak{l}_{0}$.

######  Theorem 4.1.

([DAZi, Theorem 1, p. 9]) Under the notations above a left-invariant metric on
$G$ of the form

$\langle\ ,\ \rangle=x\cdot
Q|_{\mathfrak{m}}+A_{0}|_{\mathfrak{l}_{0}}+u_{1}\cdot
Q|_{\mathfrak{l}_{1}}+\cdots+u_{p}\cdot
Q|_{\mathfrak{l}_{p}},\quad(x,u_{1},\dots,u_{p}>0)$ (13)

is naturally reductive with respect to $G\times L$, where $G\times L$ acts on
$G$ by $(g,l)y=gyl^{-1}$.

Moreover, if a left-invariant metric $\langle\ ,\ \rangle$ on a compact simple
Lie group $G$ is naturally reductive, then there is a closed subgroup $L$ of
$G$ and the metric $\langle\ ,\ \rangle$ is given by the form (13).

For the Lie group $\operatorname{SO}(n)$, we consider left-invariant metrics
determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
scalar products of the form (10) where $n=k_{1}+k_{2}+k_{3}$. Recall that
$K=\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})$
with Lie algebra
$\mathfrak{k}=\mathfrak{so}(k_{1})\oplus\mathfrak{so}(k_{2})\oplus\mathfrak{so}(k_{3})$.

######  Proposition 4.2.

If a left invariant metric $\langle\ ,\ \rangle$ of the form (10) on
$\operatorname{SO}(n)$ is naturally reductive with respect to
$\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$, then one of the following holds:

1)  $x_{1}=x_{2}=x_{12}$, $x_{13}=x_{23}$ 2) $x_{2}=x_{3}=x_{23}$,
$x_{12}=x_{13}$ 3) $x_{1}=x_{3}=x_{13}$, $x_{12}=x_{23}$, 4)
$x_{12}=x_{13}=x_{23}$.

Conversely, if one of the conditions 1), 2), 3), 4) is satisfied, then the
metric $\langle\ ,\ \rangle$ of the form (10) is naturally reductive with
respect to $\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$.

###### Proof.

Let ${\mathfrak{l}}$ be the Lie algebra of $L$. Then we have either
${\mathfrak{l}}\subset{\mathfrak{k}}$ or
${\mathfrak{l}}\not\subset{\mathfrak{k}}$. First we consider the case of
${\mathfrak{l}}\not\subset{\mathfrak{k}}$. Let ${\mathfrak{h}}$ be the
subalgebra of ${\mathfrak{g}}$ generated by ${\mathfrak{l}}$ and
${\mathfrak{k}}$. Since
$\mathfrak{so}(k_{1}+k_{2}+k_{3})=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}$
is an irreducible decomposition as $\mbox{Ad}(K)$-modules, we see that the Lie
algebra $\mathfrak{h}$ contains at least one of ${\mathfrak{m}}_{12}$,
${\mathfrak{m}}_{13}$, ${\mathfrak{m}}_{23}$. We first consider the case that
$\mathfrak{h}$ contains ${\mathfrak{m}}_{12}$. Note that
$\left[{\mathfrak{m}}_{12},{\mathfrak{m}}_{12}\right]={\mathfrak{m}}_{1}\oplus\mathfrak{m}_{2}$
and $\mathfrak{m}_{1}\oplus{\mathfrak{m}}_{2}\oplus{\mathfrak{m}}_{12}$ is a
subalgebra $\mathfrak{so}(k_{1}+k_{2})$. Thus we see that $\mathfrak{h}$
contains $\mathfrak{so}(k_{1}+k_{2})\oplus\mathfrak{so}(k_{3})$. If
$\mathfrak{h}=\mathfrak{so}(k_{1}+k_{2})\oplus\mathfrak{so}(k_{3})$, then we
obtain an irreducible decomposition
$\mathfrak{so}(k_{1}+k_{2}+k_{3})=\mathfrak{h}\oplus\mathfrak{n}$, where
$\mathfrak{n}={\mathfrak{m}}_{13}\oplus{\mathfrak{m}}_{23}$. Hence, the metric
$\langle\ ,\ \rangle$ of the form (10) satisfies $x_{1}=x_{2}=x_{12}$ and
$x_{13}=x_{23}$, so we obtain case 1). Cases 2) and 3) are obtained by a
similar way.

Now we consider the case ${\mathfrak{l}}\subset{\mathfrak{k}}$. Since the
orthogonal complement ${\mathfrak{l}}^{\bot}$ of ${\mathfrak{l}}$ with respect
to $-B$ contains the orthogonal complement ${\mathfrak{k}}^{\bot}$ of
${\mathfrak{k}}$, we see that
${\mathfrak{l}}^{\bot}\supset{\mathfrak{m}}_{12}\oplus{\mathfrak{m}}_{13}\oplus{\mathfrak{m}}_{23}$.
Since the invariant metric $\langle\ ,\ \rangle$ is naturally reductive with
respect to $G\times L$, it follows that $x_{12}=x_{13}=x_{23}$ by Theorem 4.1.
The converse is a direct consequence of Theorem 4.1. ∎

For the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2}))$-invariant
metrics of the form (11) the above proposition reduces to the following:

######  Proposition 4.3.

If a left invariant metric $\langle\ ,\ \rangle$ of the form (11) on
$\operatorname{SO}(n)$ is naturally reductive with respect to
$\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$, then one of the following holds:

1)  $x_{1}=x_{2}=x_{12}$, $x_{13}=x_{23}$, 2) $x_{2}=x_{23}$, $x_{12}=x_{13}$,
3) $x_{1}=x_{13}$, $x_{12}=x_{23}$ , 4) $x_{12}=x_{13}=x_{23}$.

Conversely, if one of the conditions 1), 2), 3), 4) is satisfied, then the
metric $\langle\ ,\ \rangle$ of the form (11) is naturally reductive with
respect to $\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$.

Finally, for the $\operatorname{Ad}(\operatorname{SO}(k_{1}))$-invariant
metrics of the form (12) we have the following:

######  Proposition 4.4.

If a left invariant metric $\langle\ ,\ \rangle$ of the form (12) on
$\operatorname{SO}(n)$ is naturally reductive with respect to
$\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$, then one of the following holds:

1)  $x_{1}=x_{12}$, $x_{13}=x_{23}$, 2) $x_{1}=x_{13}$, $x_{12}=x_{23}$, 3)
$x_{12}=x_{13}$.

Conversely, if one of the conditions 1), 2), 3) is satisfied, then the metric
$\langle\ ,\ \rangle$ of the form (12) is naturally reductive with respect to
$\operatorname{SO}(n)\times L$ for some closed subgroup $L$ of
$\operatorname{SO}(n)$.

##  5\. The Ricci tensor for a class of left-invariant metrics on
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$

We will compute the Ricci tensor for the left-invariant metrics on
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$, determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
scalar products of the form (10). We use Lemma 2.1 taking into account (9),
and we obtain the following:

######  Proposition 5.1.

The components of the Ricci tensor ${r}$ for the left-invariant metric
$\langle\ ,\ \rangle$ on $G$ defined by (10) are given as follows:

$\small{\begin{array}[]{lll}r_{1}&=&\displaystyle{\frac{1}{2x_{1}}+\frac{1}{4d_{1}}\biggl{(}{1\brack{11}}\frac{1}{x_{1}}+{1\brack{(12)(12)}}\frac{x_{1}}{{x_{12}}^{2}}+{1\brack{(13)(13)}}\frac{x_{1}}{{x_{13}}^{2}}\biggr{)}}\\\
\\\
&&\displaystyle{-\frac{1}{2d_{1}}\biggl{(}{1\brack{11}}\frac{1}{x_{1}}+{(12)\brack{1(12)}}\frac{1}{{x_{1}}}+{(13)\brack{1(13)}}\frac{1}{{x_{1}}}\biggr{)},}\\\
\\\
r_{2}&=&\displaystyle{\frac{1}{2x_{2}}+\frac{1}{4d_{2}}\biggl{(}{2\brack{22}}\frac{1}{x_{2}}+{2\brack{(12)(12)}}\frac{x_{2}}{{x_{12}}^{2}}+{2\brack{(23)(23)}}\frac{x_{2}}{{x_{23}}^{2}}\biggr{)}}\\\
\\\
&&\displaystyle{-\frac{1}{2d_{2}}\biggl{(}{2\brack{22}}\frac{1}{x_{2}}+{(12)\brack{2(12)}}\frac{1}{{x_{2}}}+{(23)\brack{2(23)}}\frac{1}{{x_{2}}}\biggr{)},}\\\
\\\
r_{3}&=&\displaystyle{\frac{1}{2x_{3}}+\frac{1}{4d_{3}}\biggl{(}{3\brack{33}}\frac{1}{x_{3}}+{3\brack{(13)(13)}}\frac{x_{3}}{{x_{13}}^{2}}+{3\brack{(23)(23)}}\frac{x_{3}}{{x_{23}}^{2}}\biggr{)}}\\\
\\\
&&\displaystyle{-\frac{1}{2d_{3}}\biggl{(}{3\brack{33}}\frac{1}{x_{3}}+{(13)\brack{3(13)}}\frac{1}{{x_{3}}}+{(23)\brack{3(23)}}\frac{1}{{x_{3}}}\biggr{)},}\\\
\\\
r_{12}&=&\displaystyle{\frac{1}{2x_{12}}+\frac{1}{4d_{12}}\biggl{(}{(12)\brack{1(12)}}\frac{1}{x_{1}}\times
2+{(12)\brack{2(12)}}\frac{1}{x_{2}}\times
2+{(12)\brack{(13)(23)}}\frac{x_{12}}{x_{13}x_{23}}\times 2\biggr{)}}\\\ \\\
&&\displaystyle{-\frac{1}{2d_{12}}\biggl{(}{1\brack{(12)(12)}}\frac{x_{1}}{{x_{12}}^{2}}+{(12)\brack{(12)1}}\frac{1}{x_{1}}+{2\brack{(12)(12)}}\frac{x_{2}}{{x_{12}}^{2}}+{(12)\brack{(12)2}}\frac{1}{x_{2}}}\\\
\\\
&&\displaystyle{+{(13)\brack{(12)(23)}}\frac{x_{13}}{x_{12}x_{23}}+{(23)\brack{(12)(13)}}\frac{x_{23}}{x_{12}x_{13}}\biggr{)}},\\\
\\\
r_{13}&=&\displaystyle{\frac{1}{2x_{13}}+\frac{1}{4d_{13}}\biggl{(}{(13)\brack{1(13)}}\frac{1}{x_{1}}\times
2+{(13)\brack{2(13)}}\frac{1}{x_{2}}\times
2+{(13)\brack{(12)(23)}}\frac{x_{13}}{x_{12}x_{23}}\times 2\biggr{)}}\\\ \\\
&&\displaystyle{-\frac{1}{2d_{13}}\biggl{(}{1\brack{(13)(13)}}\frac{x_{1}}{{x_{13}}^{2}}+{(13)\brack{(13)1}}\frac{1}{x_{1}}+{3\brack{(13)(13)}}\frac{x_{3}}{{x_{13}}^{2}}+{(13)\brack{(13)3}}\frac{1}{x_{3}}}\\\
\\\
&&\displaystyle{+{(12)\brack{(13)(23)}}\frac{x_{12}}{x_{13}x_{23}}+{(23)\brack{(13)(12)}}\frac{x_{23}}{x_{13}x_{12}}\biggr{)}},\end{array}}$
(14) $\small{\begin{array}[]{lll}\\\ \\\
r_{23}&=&\displaystyle{\frac{1}{2x_{23}}+\frac{1}{4d_{23}}\biggl{(}{(23)\brack{2(23)}}\frac{1}{x_{2}}\times
2+{(23)\brack{3(23)}}\frac{1}{x_{3}}\times
2+{(23)\brack{(12)(13)}}\frac{x_{23}}{x_{12}x_{13}}\times 2\biggr{)}}\\\ \\\
&&\displaystyle{-\frac{1}{2d_{23}}\biggl{(}{2\brack{(23)(23)}}\frac{x_{2}}{{x_{23}}^{2}}+{(23)\brack{(23)2}}\frac{1}{x_{2}}+{3\brack{(23)(23)}}\frac{x_{3}}{{x_{23}}^{2}}+{(23)\brack{(23)3}}\frac{1}{x_{3}}}\\\
\\\
&&\displaystyle{+{(12)\brack{(23)(13)}}\frac{x_{12}}{x_{23}x_{13}}+{(13)\brack{(23)(12)}}\frac{x_{13}}{x_{23}x_{12}}\biggr{)}},\end{array}}$

where $n=k_{1}+k_{2}+k_{3}$.

We recall the following lemma from [ArDzNi] (a detailed proof was given in
[ArSaSt1]).

######  Lemma 5.2.

([ArDzNi, Lemma 4.2]) For $a,b,c=1,2,3$ and $(a-b)(b-c)(c-a)\neq 0$ the
following relations hold:

$\begin{array}[]{lll}\displaystyle{{a\brack{aa}}=\frac{k_{a}(k_{a}-1)(k_{a}-2)}{2(n-2)}},&\displaystyle{{a\brack{(ab)(ab)}}=\frac{k_{a}k_{b}(k_{a}-1)}{2(n-2)}},&\displaystyle{{(ac)\brack{(ab)(bc)}}=\frac{k_{a}k_{b}k_{c}}{2(n-2)}}.\end{array}$
(15)

By using the above lemma, we can now obtain the components of the Ricci tensor
for the metrics we are considering in this work.

######  Proposition 5.3.

The components of the Ricci tensor ${r}$ for the left-invariant metric
$\langle\ ,\ \rangle$ on $G$ defined by (10) are given as follows:

$\left.\small{\begin{array}[]{l}r_{1}=\displaystyle{\frac{k_{1}-2}{4(n-2)x_{1}}+\frac{1}{4(n-2)}\biggl{(}k_{2}\frac{x_{1}}{{x_{12}}^{2}}}+k_{3}\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\\\
\\\
r_{2}=\displaystyle{\frac{k_{2}-2}{4(n-2)x_{2}}+\frac{1}{4(n-2)}\biggl{(}k_{1}\frac{x_{2}}{{x_{12}}^{2}}+k_{3}\frac{x_{2}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{3}=\displaystyle{\frac{k_{3}-2}{4(n-2)x_{3}}+\frac{1}{4(n-2)}\biggl{(}k_{1}\frac{x_{3}}{{x_{13}}^{2}}+k_{2}\frac{x_{3}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{12}=\displaystyle{\frac{1}{2x_{12}}+\frac{k_{3}}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}\displaystyle{-\frac{1}{4(n-2)}\biggl{(}\frac{(k_{1}-1)x_{1}}{{x_{12}}^{2}}+\frac{(k_{2}-1)x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{13}=\displaystyle{\frac{1}{2x_{13}}+\frac{k_{2}}{4(n-2)}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}\displaystyle{-\frac{1}{4(n-2)}\biggl{(}\frac{(k_{1}-1)x_{1}}{{x_{13}}^{2}}+\frac{(k_{3}-1)x_{3}}{{x_{13}}^{2}}\biggr{)}},\\\
\\\
r_{23}=\displaystyle{\frac{1}{2x_{23}}+\frac{k_{1}}{4(n-2)}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}}\displaystyle{-\frac{1}{4(n-2)}\biggl{(}\frac{(k_{2}-1)x_{2}}{{x_{23}}^{2}}+\frac{(k_{3}-1)x_{3}}{{x_{23}}^{2}}\biggr{)}},\end{array}}\right\\}$
(16)

where $n=k_{1}+k_{2}+k_{3}$.

For the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2}))$-invariant
metrics on $G$ of the form (11), Proposition 5.3 reduces to

######  Proposition 5.4.

The components of the Ricci tensor ${r}$ for the left-invariant metric
$\langle\ ,\ \rangle$ on $G$ defined by (11) are given as follows:

$\left.\small{\begin{array}[]{l}r_{1}=\displaystyle{\frac{k_{1}-2}{4(n-2)x_{1}}+\frac{1}{4(n-2)}\biggl{(}k_{2}\frac{x_{1}}{{x_{12}}^{2}}}+\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\\\
\\\
r_{2}=\displaystyle{\frac{k_{2}-2}{4(n-2)x_{2}}+\frac{1}{4(n-2)}\biggl{(}k_{1}\frac{x_{2}}{{x_{12}}^{2}}+\frac{x_{2}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{12}=\displaystyle{\frac{1}{2x_{12}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}-\frac{(k_{1}-1)x_{1}}{{x_{12}}^{2}}-\frac{(k_{2}-1)x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{13}=\displaystyle{\frac{1}{2x_{13}}+\frac{1}{4(n-2)}\biggl{(}{k_{2}}\left(\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\right)-\frac{(k_{1}-1)x_{1}}{{x_{13}}^{2}}\biggr{)}},\\\
\\\
r_{23}=\displaystyle{\frac{1}{2x_{23}}+\frac{1}{4(n-2)}\biggl{(}k_{1}\left(\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\right)-\frac{(k_{2}-1)x_{2}}{{x_{23}}^{2}}\biggr{)}},\end{array}}\right\\}$
(17)

where $n=k_{1}+k_{2}+1$.

Finally, for the $\operatorname{Ad}(\operatorname{SO}(k_{1}))$-invariant
metrics on $G$ of the form (12), Proposition 5.3 reduces to

######  Proposition 5.5.

The components of the Ricci tensor ${r}$ for the left-invariant metric
$\langle\ \ ,\ \ \rangle$ on $G$ defined by (12) are given as follows:

$\left.\small{\begin{array}[]{l}r_{1}=\displaystyle{\frac{n-4}{4(n-2)x_{1}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{1}}{{x_{12}}^{2}}}+\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\\\
\\\
r_{12}=\displaystyle{\frac{1}{2x_{12}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}-\frac{(n-3)x_{1}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{13}=\displaystyle{\frac{1}{2x_{13}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}-\frac{(n-3)x_{1}}{{x_{13}}^{2}}\biggr{)}},\\\
\\\
r_{23}=\displaystyle{\frac{1}{2x_{23}}+\frac{1}{4}\left(\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\right)},\end{array}}\right\\}$
(18)

where $n=k_{1}+2$.

##  6\. Left-invariant Einstein metrics on $\operatorname{SO}(n)$

In the present section we provide detailed proofs on how to obtain left-
invariant Einstein metrics which are not naturally reductive for the Lie
groups $\operatorname{SO}(7)$, $\operatorname{SO}(8)$ and
$\operatorname{SO}(n)$, $n\geq 9$. For $\operatorname{SO}(7)$ and
$\operatorname{SO}(8)$ we also describe left-invariant Einstein metrics which
are naturally reductive. The other compact Lie groups
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$ for $n\geq 7$ and
for all possible
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
metrics of the forms (10) and (11) can be treated in an analogous manner and
we omit the proofs. We summarise all the results at the end of Section 7.
Here, we provide information about solving or proving existence of solutions
of algebraic systems of equations. These solutions correspond to Einstein
metrics which are not naturally reductive.

######  Proposition 6.1.

The Lie group $\operatorname{SO}(7)$ admits at least one left-invariant
Einstein metric determined by the
$\operatorname{Ad}(\operatorname{SO}(3)\times\operatorname{SO}(3))$-invariant
scalar product of the form (11), which is not naturally reductive.

###### Proof.

This is the case when $k_{1}=k_{2}=3$ and $k_{3}=1$. From Proposition 5.4, we
see that the components of the Ricci tensor ${r}$ for the invariant metric are
given by

$\left.\small{\begin{array}[]{lll}r_{1}&=&\displaystyle{\frac{1}{20x_{1}}+\frac{1}{20}\biggl{(}3\frac{x_{1}}{{x_{12}}^{2}}}+\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\quad
r_{2}\,\,\,=\,\,\,\displaystyle{\frac{1}{20x_{2}}+\frac{1}{20}\biggl{(}3\frac{x_{2}}{{x_{12}}^{2}}+\frac{x_{2}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{12}&=&\displaystyle{\frac{1}{2x_{12}}+\frac{1}{20}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}\displaystyle{-\frac{1}{10}\biggl{(}\frac{x_{1}}{{x_{12}}^{2}}+\frac{x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{23}&=&\displaystyle{\frac{1}{2x_{23}}+\frac{3}{20}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}-\frac{1}{10}\frac{x_{2}}{{x_{23}}^{2}}},\\\
\\\
r_{13}&=&\displaystyle{\frac{1}{2x_{13}}+\frac{3}{20}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}-\frac{1}{10}\frac{x_{1}}{{x_{13}}^{2}}}.\end{array}}\right\\}$
(19)

We consider the system of equations

$r_{1}=r_{2},\,\,r_{2}=r_{12},\,\,r_{12}=r_{23},\,\,r_{23}=r_{13}.$ (20)

Then finding Einstein metrics of the form (11) reduces to finding the positive
solutions of system (20), and we normalize our equations by putting
$x_{23}=1$. Then we obtain the system of equations:

$\left.{\begin{array}[]{l}g_{1}={x_{1}}^{2}{x_{12}}^{2}{x_{2}}+3{x_{1}}^{2}{x_{13}}^{2}{x_{2}}-{x_{1}}{x_{12}}^{2}{x_{13}}^{2}{x_{2}}^{2}-{x_{1}}{x_{12}}^{2}{x_{13}}^{2}\\\
\quad\quad-3{x_{1}}{x_{13}}^{2}{x_{2}}^{2}+{x_{12}}^{2}{x_{13}}^{2}{x_{2}}=0,\\\
g_{2}=2{x_{1}}{x_{13}}{x_{2}}-{x_{12}}^{3}{x_{2}}+{x_{12}}^{2}{x_{13}}{x_{2}}^{2}+{x_{12}}^{2}{x_{13}}+{x_{12}}{x_{13}}^{2}{x_{2}}\\\
\quad\quad-10{x_{12}}{x_{13}}{x_{2}}+{x_{12}}{x_{2}}+5{x_{13}}{x_{2}}^{2}=0,\\\
g_{3}=-{x_{1}}{x_{13}}+2{x_{12}}^{3}+{x_{12}}^{2}{x_{13}}{x_{2}}-5{x_{12}}^{2}{x_{13}}+{x_{12}}{x_{13}}^{2}\\\
\quad\quad+5{x_{12}}{x_{13}}-2{x_{12}}-{x_{13}}{x_{2}}=0,\\\
g_{4}={x_{1}}{x_{12}}-{x_{12}}{x_{13}}^{2}{x_{2}}+5{x_{12}}{x_{13}}^{2}-5{x_{12}}{x_{13}}-3{x_{13}}^{3}+3{x_{13}}=0.\end{array}}\right\\}$
(21)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$
and an ideal $I$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1\\}$ to find non zero
solutions of equations (21). We take a lexicographic order $>$ with
$z>x_{1}>x_{2}>x_{12}>x_{13}$ for a monomial ordering on $R$. Then, by the aid
of computer, we see that a Gröbner basis for the ideal $I$ contains the
polynomial

$({x_{13}}-1)\left(6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45\right)\left(45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6\right)\,h_{1}(x_{13}),$

where $h_{1}(x_{13})$ is a polynomial of $x_{13}$ given by

$\displaystyle
h_{1}(x_{13})=9078544800000{x_{13}}^{24}-87978150000000{x_{13}}^{23}+416122213455000{x_{13}}^{22}$
$\displaystyle-1222223075437500{x_{13}}^{21}+2532878590309970{x_{13}}^{20}-4171390831990050{x_{13}}^{19}$
$\displaystyle+5900094406718764{x_{13}}^{18}-7070644584919459{x_{13}}^{17}+6230617318198202{x_{13}}^{16}$
$\displaystyle-4091340309226802{x_{13}}^{15}+1722695469975774{x_{13}}^{14}+983550542994755{x_{13}}^{13}$
$\displaystyle-2624020500593532{x_{13}}^{12}+983550542994755{x_{13}}^{11}+1722695469975774{x_{13}}^{10}$
$\displaystyle-4091340309226802{x_{13}}^{9}+6230617318198202{x_{13}}^{8}-7070644584919459{x_{13}}^{7}$
$\displaystyle+5900094406718764{x_{13}}^{6}-4171390831990050{x_{13}}^{5}+2532878590309970{x_{13}}^{4}$
$\displaystyle-1222223075437500{x_{13}}^{3}+416122213455000{x_{13}}^{2}-87978150000000{x_{13}}$
$\displaystyle+9078544800000.$

By solving the equation $h_{1}(x_{13})=0$ numerically, we obtain two positive
solutions $x_{13}=a_{13}$ and $x_{13}=b_{13}$ which are given approximately as
$a_{13}\approx 0.4254295,\,\,b_{13}\approx 2.350565.$ We also see that the
Gröbner basis for the ideal $I$ contains the polynomials

$x_{12}-w_{12}(x_{13}),\quad x_{1}-w_{1}(x_{13}),\quad x_{2}-w_{2}(x_{13}),$

where $w_{12}(x_{13})$, $w_{1}(x_{13})$ and $w_{2}(x_{13})$ are polynomials of
$x_{13}$ with rational coefficients. By substituting the values $a_{13}$ and
$b_{13}$ for $x_{13}$ into $w_{12}(x_{13})$, $w_{1}(x_{13})$ and
$w_{2}(x_{13})$, we obtain two positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0\\}$ approximately as

$(x_{13},x_{12},x_{1},x_{2})\approx(0.4254295,\,0.9312204,\,0.1200109,\,0.1122291),$
$(x_{13},x_{12},x_{1},x_{2})\approx(2.350565,\,2.188895,\,0.2638018,\,0.2820935).$

We substitute these values into the system (19) together with $x_{23}=1$. Then
we obtain that $r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.470542$ and
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.200182$ respectively. We multiply
these solutions by a scale factor and we obtain the two solutions

$(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.0564701,\,0.0528085,\,0.438178,\,0.470542,\,0.20018),$
$(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.0528085,\,0.0564701,\,0.438178,\,0.20018,\,0.470542).$

for the system of equations

$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}=1,$

so we see that these two solutions are isometric. Note that this metric is not
naturally reductive from Proposition 4.3.

Now we consider the case

$({x_{13}}-1)\left(6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45\right)\left(45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6\right)=0.$

We consider the ideals $J_{1}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45\\}$,
$J_{2}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6\\}$
and $J_{3}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,{x_{13}}-1\\}$ of the
polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$.

We take a lexicographic order $>$ with $z>x_{1}>x_{2}>x_{12}>x_{13}$ for a
monomial ordering on $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$. Then, by
the aid of computer, we see that a Gröbner basis for the ideal $J_{1}$ is
given by

$\displaystyle\\{6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45,x_{12}-{x_{13}},x_{2}-1,x_{1}+{x_{13}}^{2}-5{x_{13}}+3,$
$\displaystyle-804{x_{13}}^{2}+5284{x_{13}}+405z-8112\\}.$

By solving the equation $6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45=0$
numerically, we obtain three positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,x_{13}-1,6{x_{13}}^{3}-44{x_{13}}^{2}+90{x_{13}}-45=0\\}$
approximately as

$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(4.16278,\,4.16278,\,0.485171,\,1,\,1),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(2.42874,\,2.42874,\,3.24492,\,1,\,1),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.741818,\,0.741818,\,0.158797,\,1,\,1).$

We substitute these values into the system (19). Then we obtain that
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.108656$,
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.125429$ and
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.372581$ respectively. We multiply
these solutions by a scale factor and we obtain three solutions

$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.452311,\,0.452311,\,0.0527168,\,0.108656,\,0.108656),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.304634,\,0.304634,\,0.407007,\,0.125429,\,0.125429),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.276388,\,0.276388,\,0.0591647,\,0.372581,\,0.372581).$

for the system of equations

$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}=1,$

We also see that a Gröbner basis for the ideal $J_{2}$ is given by

$\displaystyle\\{45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6,x_{12}-1,45{x_{13}}^{2}-72{x_{13}}+6{x_{2}}+14,{x_{1}}-{x_{13}},$
$\displaystyle-4545{x_{13}}^{2}+8010{x_{13}}+6z-2554\\}.$

By solving the equation $45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6=0$
numerically, we obtain three positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,x_{13}-1,45{x_{13}}^{3}-90{x_{13}}^{2}+44{x_{13}}-6=0\\}$
approximately as

$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.240224,\,1,\,0.240224,\,0.11655,\,1),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(0.411737,\,1,\,0.411737,\,1.33605,\,1),$
$\displaystyle(x_{13},x_{12},x_{1},x_{2},x_{23})\approx(1.34804,\,1,\,1.34804,\,0.214064,\,1).$

We substitute these values into the system (19). Then we obtain that
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.452311$,
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.304634$ and
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.276388$ respectively. We multiply
these solutions by a scale factor and we obtain three solutions

$\displaystyle(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.108656,\,0.0527168,\,0.452311,\,0.452311,\,0.108656),$
$\displaystyle(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.125429,\,0.407007,\,0.304634,\,0.304634,\,0.125429),$
$\displaystyle(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.372581,\,0.0591647,\,0.276388,\,0.276388,\,0.372581).$

for the system of equations

$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}=1.$

Note that, from these two cases, we obtain three Einstein metrics up to
isometry.

Now we consider the case of ideal $J_{3}$. We see that a Gröbner basis for the
ideal $J_{3}$ is given by

$\displaystyle\\{{x_{13}}-1,({x_{12}}-1)(3{x_{12}}-2)\left(2{x_{12}}^{4}-5{x_{12}}^{3}+19{x_{12}}^{2}-35{x_{12}}+26\right),$
$\displaystyle-24{x_{12}}^{5}+70{x_{12}}^{4}-267{x_{12}}^{3}+550{x_{12}}^{2}-522{x_{12}}+63{x_{2}}+130,$
$\displaystyle
63{x_{1}}-24{x_{12}}^{5}+70{x_{12}}^{4}-267{x_{12}}^{3}+550{x_{12}}^{2}-522{x_{12}}+130,$
$\displaystyle-569934{x_{12}}^{5}+2980005{x_{12}}^{4}-8034374{x_{12}}^{3}+22826670{x_{12}}^{2}$
$\displaystyle-25019231{x_{12}}+246064z+7570800\\}.$

By solving the equation
$2{x_{12}}^{4}-5{x_{12}}^{3}+19{x_{12}}^{2}-35{x_{12}}+26=0$ numerically, we
see that there are no real solutions. For $x_{12}=1$, we obtain that
$x_{1}=x_{2}=x_{13}=x_{23}=1$, that is, the metric is bi-invariant. For
$x_{12}=2/3$, we obtain that $x_{1}=x_{2}=2/3$ and $x_{13}=x_{23}=1$, hence we
obtain two Einstein metrics up to isometry.

From Proposition 4.3 it follows that the five above metrics obtained are all
naturally reductive with respect to $\operatorname{SO}(7)\times L$, where $L$
is a closed subgroup of $\operatorname{SO}(7)$, which is either
$\operatorname{SO}(3)\times\operatorname{SO}(4)$, $\operatorname{SO}(6)$ or
$\operatorname{SO}(7)$. ∎

######  Proposition 6.2.

The Lie group $\operatorname{SO}(8)$ admits at least two (non isometric) left-
invariant Einstein metrics determined by the
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(3))$-invariant
scalar products of the form (11), which are not naturally reductive.

###### Proof.

This is the case when $k_{1}=4,k_{2}=3$ and $k_{3}=1$. From Proposition 5.4,
we see that the components of the Ricci tensor ${r}$ for the invariant metric
are given by

$\left.\small{\begin{array}[]{lll}r_{1}&=&\displaystyle{\frac{1}{12x_{1}}+\frac{1}{24}\biggl{(}3\frac{x_{1}}{{x_{12}}^{2}}}+\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\quad
r_{2}\,\,\,=\,\,\,\displaystyle{\frac{1}{24x_{2}}+\frac{1}{24}\biggl{(}4\frac{x_{2}}{{x_{12}}^{2}}+\frac{x_{2}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{12}&=&\displaystyle{\frac{1}{2x_{12}}+\frac{1}{24}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}\displaystyle{-\frac{1}{24}\biggl{(}3\frac{x_{1}}{{x_{12}}^{2}}+2\frac{x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{23}&=&\displaystyle{\frac{1}{2x_{23}}+\frac{1}{6}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}-\frac{1}{12}\frac{x_{2}}{{x_{23}}^{2}}},\\\
\\\
r_{13}&=&\displaystyle{\frac{1}{2x_{13}}+\frac{1}{8}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}-\frac{1}{8}\frac{x_{1}}{{x_{13}}^{2}}}.\par\end{array}}\right\\}$
(22)

We consider the system of equations

$r_{1}=r_{2},\,\,r_{2}=r_{12},\,\,r_{12}=r_{23},\,\,r_{23}=r_{13}.$ (23)

Then finding Einstein metrics of the form (11) reduces to finding the positive
solutions of system (23), and we normalize our equations by putting
$x_{23}=1$. Then we have the system of equations:

$\left.{\begin{array}[]{l}g_{1}={x_{1}}^{2}{x_{12}}^{2}{x_{2}}+3{x_{1}}^{2}{x_{13}}^{2}{x_{2}}-{x_{1}}{x_{12}}^{2}{x_{13}}^{2}{x_{2}}^{2}-{x_{1}}{x_{12}}^{2}{x_{13}}^{2}\\\
\quad\quad-4{x_{1}}{x_{13}}^{2}{x_{2}}^{2}+2{x_{12}}^{2}{x_{13}}^{2}{x_{2}}=0,\\\
g_{2}=3{x_{1}}{x_{13}}{x_{2}}-{x_{12}}^{3}{x_{2}}+{x_{12}}^{2}{x_{13}}{x_{2}}^{2}+{x_{12}}^{2}{x_{13}}+{x_{12}}{x_{13}}^{2}{x_{2}}\\\
\quad\quad-12{x_{12}}{x_{13}}{x_{2}}+{x_{12}}{x_{2}}+6{x_{13}}{x_{2}}^{2}=0,\\\
g_{3}=-3{x_{1}}{x_{13}}+5{x_{12}}^{3}+2{x_{12}}^{2}{x_{13}}{x_{2}}-12{x_{12}}^{2}{x_{13}}+3{x_{12}}{x_{13}}^{2}\\\
\quad\quad+12{x_{12}}{x_{13}}-5{x_{12}}-2{x_{13}}{x_{2}}=0,\\\
g_{4}=3{x_{1}}{x_{12}}-{x_{12}}^{2}{x_{13}}-2{x_{12}}{x_{13}}^{2}{x_{2}}+12{x_{12}}{x_{13}}^{2}-12{x_{12}}{x_{13}}\\\
\quad\quad-7{x_{13}}^{3}+7{x_{13}}=0.\end{array}}\right\\}$ (24)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$
and an ideal $I$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1\\}$ to find non zero
solutions of equations (24). We take a lexicographic order $>$ with
$z>x_{1}>x_{2}>x_{12}>x_{13}$ for a monomial ordering on $R$. Then, by the aid
of computer, we see that a Gröbner basis for the ideal $I$ contains the
polynomial

$({x_{13}}-5)({x_{13}}-1)\left(7{x_{13}}^{2}-24{x_{13}}+14\right)\left(287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63\right)\,h_{2}(x_{13}),$

where $h_{2}(x_{13})$ is a polynomial of $x_{13}$ given by

$\displaystyle
h_{2}(x_{13})=5426775507148489670400{x_{13}}^{28}-85161185092622977873920{x_{13}}^{27}$
$\displaystyle+643415930216926223949312{x_{13}}^{26}-3054548385819855899001216{x_{13}}^{25}$
$\displaystyle+10179140499777121100664800{x_{13}}^{24}-25585147362416655835236384{x_{13}}^{23}$
$\displaystyle+51380426324079059150364272{x_{13}}^{22}-85934185504663087173249048{x_{13}}^{21}$
$\displaystyle+120352447918421302289568863{x_{13}}^{20}-136938372384910964649260802{x_{13}}^{19}$
$\displaystyle+121268417379459335461167457{x_{13}}^{18}-78483773118912467818333590{x_{13}}^{17}$
$\displaystyle+32048679980888195807658286{x_{13}}^{16}-21037081214018592447662850{x_{13}}^{15}$
$\displaystyle+96567724403906545251348604{x_{13}}^{14}-279673822213789859470643520{x_{13}}^{13}$
$\displaystyle+527833035046902978479331387{x_{13}}^{12}-769632045866390647274523642{x_{13}}^{11}$
$\displaystyle+937521733316934021780397473{x_{13}}^{10}-973318915329328329165562374{x_{13}}^{9}$
$\displaystyle+864907599634224063462448416{x_{13}}^{8}-664413545084655303518836950{x_{13}}^{7}$
$\displaystyle+442175543674339070418041970{x_{13}}^{6}-249282932584983174857359764{x_{13}}^{5}$
$\displaystyle+114233981412525395978707920{x_{13}}^{4}-40474281023127469650239100{x_{13}}^{3}$
$\displaystyle+10382320721058779134026000{x_{13}}^{2}-1735984447231701886065000{x_{13}}$
$\displaystyle+146138820428187141975000.$

By solving the equation $h_{2}(x_{13})=0$ numerically, we obtain two positive
solutions $x_{13}=a_{13}$ and $x_{13}=b_{13}$ which are given approximately as
$a_{13}\approx 0.48183112,\,\,b_{13}\approx 2.7966957.$ We also see that the
Gröbner basis for the ideal $I$ contains the polynomials

$x_{12}-w_{12}(x_{13}),\quad x_{1}-w_{1}(x_{13}),\quad x_{2}-w_{2}(x_{13}),$

where $w_{12}(x_{13})$, $w_{1}(x_{13})$ and $w_{2}(x_{13})$ are polynomials of
$x_{13}$ with rational coefficients. By substituting the values $a_{13}$ and
$b_{13}$ for $x_{13}$ into $w_{12}(x_{13})$, $w_{1}(x_{13})$ and
$w_{2}(x_{13})$, we obtain two positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0\\}$ approximately as

$(x_{13},x_{12},x_{1},x_{2})\approx(0.48183112,0.90692827,0.20686292,0.092856189),$
$(x_{13},x_{12},x_{1},x_{2})\approx(2.7966957,2.6698577,0.54677135,0.28461374).$

We substitute these values into the system (22) together with $x_{23}=1$. Then
we obtain that $r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.47140698$ and
$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}\approx 0.16491085$ respectively. We multiply
these solutions by a scale factor and we obtain the two solutions

$(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.097516624,\,0.043773055,\,0.42753231,\,0.47140698,\,0.22713855),$
$(x_{1},x_{2},x_{12},x_{23},x_{13})\approx(0.090168527,\,0.046935893,\,0.44028850,\,0.16491085,\,0.46120545).$

for the system of equations

$r_{1}=r_{2}=r_{12}=r_{23}=r_{13}=1,$

which are not isometric. Note that due to Proposition 4.3. these metrics are
not naturally reductive.

Now we consider the case

$({x_{13}}-5)({x_{13}}-1)\left(7{x_{13}}^{2}-24{x_{13}}+14\right)\left(287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63\right)=0.$

We consider ideals $J_{1}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63\\}$,
$J_{2}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,7{x_{13}}^{2}-24{x_{13}}+14\\}$,
$J_{3}$ generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,{x_{13}}-5\\}$ and $J_{4}$
generated by $\\{g_{1},\,g_{2},$
$g_{3},g_{4},\,z\,x_{1}\,x_{2}\,x_{12}\,x_{13}-1,{x_{13}}-1\\}$ of the
polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$.

We take a lexicographic order $>$ with $z>x_{1}>x_{2}>x_{12}>x_{13}$ for a
monomial ordering on $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$. Then, by
the aid of computer, we see that a Gröbner basis for the ideal $J_{1}$ is
given by

$\displaystyle\\{287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63,x_{12}-1,117-478x_{13}+287{x_{13}}^{2}+42x_{2},x_{1}-{x_{13}},$
$\displaystyle-123201+323882x_{13}-173635{x_{13}}^{2}+378z\\}.$

By solving the equation $287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63=0$
numerically, we obtain three positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,x_{13}-1,287{x_{13}}^{3}-625{x_{13}}^{2}+369{x_{13}}-63=0\\}$.
Since the solutions satisfy $x_{1}=x_{13}$, $x_{12}=x_{23}=1$, then
Proposition 4.3 implies that the metrics obtained are naturally reductive with
respect to
$\operatorname{SO}(8)\times(\operatorname{SO}(5)\times\operatorname{SO}(3))$.

Similarly, we see that a Gröbner basis for the ideal $J_{2}$ is given by

$\displaystyle\\{14-24x_{13}+7{x_{13}}^{2},x_{12}-x_{13},-1+x_{2},7+7x_{1}-12x_{13},-43009+15960x_{13}+4802z\\}.$

By solving the equation $14-24x_{13}+7{x_{13}}^{2}=0$, we obtain two positive
solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,14-24x_{13}+7{x_{13}}^{2}=0\\}$. Since the
solutions satisfy $x_{2}=x_{23}=1$, $x_{12}=x_{13}=1$, then Proposition 4.3
implies that the metrics obtained are naturally reductive with respect to
$\operatorname{SO}(8)\times(\operatorname{SO}(4)\times\operatorname{SO}(4))$.
(It is possible to check that these two metrics are isometric).

Similarly, we see that a Gröbner basis for the ideal $J_{3}$ is given by

$\displaystyle\\{-5+x_{13},-5+x_{12},-1+x_{2},-1+x_{1},-1+25z\\}$

and we obtain a unique positive solution of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,x_{13}-1,-5+x_{13}=0\\}$. From Proposition
4.3 we see that the metric obtained is naturally reductive with respect to
$\operatorname{SO}(8)\times(\operatorname{SO}(4)\times\operatorname{SO}(4))$.

Finally, we see that a Gröbner basis for the ideal $J_{4}$ is given by

$\displaystyle\\{-1+x_{13},(-1+x_{12})(-5+7x_{12}),-x_{12}+x_{2},x_{1}-x_{12},-888+763x_{12}+125z\\},$

and we obtain two positive solutions of the system of equations
$\\{g_{1}=0,g_{2}=0,g_{3}=0,g_{4}=0,x_{13}-1=0\\}$. One of the solutions gives
the bi-invariant metric $x_{1}=x_{2}=x_{12}=x_{13}=x_{23}=1$ and the other
$x_{1}=x_{2}=x_{12}=5/7,x_{13}=x_{23}=1$ gives naturally reductive metric with
respect to $\operatorname{SO}(8)\times\operatorname{SO}(7)$ from Proposition
4.3. ∎

######  Proposition 6.3.

For any $n\geq 9$, the Lie group $\operatorname{SO}(n)$ admits at least one
left-invariant Einstein metric determined by the
$\operatorname{Ad}(\operatorname{SO}(n-6)\times\operatorname{SO}(3)\times\operatorname{SO}(3))$-invariant
scalar product of the form (10), which is not naturally reductive.

###### Proof.

We consider the system of equations

$r_{1}=r_{2},\quad r_{2}=r_{3},\quad r_{3}=r_{12},\quad r_{12}=r_{13},\quad
r_{13}=r_{23}.$ (25)

Then finding Einstein metrics of the form (10) reduces to finding positive
solutions of system (25).

We put $k_{2}=k_{3}=3$ and consider our equations by putting

$x_{12}=x_{13}=1,\quad x_{2}=x_{3}.$

Then the system of equations (25) reduces to the system of equations:

$\left.{\begin{array}[]{lll}g_{1}&=&-n{x_{1}}{x_{2}}^{2}{x_{23}}^{2}+n{x_{2}}{x_{23}}^{2}+6{x_{1}}^{2}{x_{2}}{x_{23}}^{2}\\\
&&+6{x_{1}}{x_{2}}^{2}{x_{23}}^{2}-3{x_{1}}{x_{2}}^{2}-{x_{1}}{x_{23}}^{2}-8{x_{2}}{x_{23}}^{2}=0,\\\
g_{2}&=&n{x_{1}}{x_{2}}{x_{23}}^{2}+n{x_{2}}^{2}{x_{23}}^{2}-2n{x_{2}}{x_{23}}^{2}-7{x_{1}}{x_{2}}{x_{23}}^{2}\\\
&&-4{x_{2}}^{2}{x_{23}}^{2}+3{x_{2}}^{2}+3{x_{2}}{x_{23}}^{3}+4{x_{2}}{x_{23}}^{2}+{x_{23}}^{2}=0,\\\
g_{3}&=&-n{x_{1}}{x_{23}}^{2}-n{x_{23}}^{3}+2n{x_{23}}^{2}+7{x_{1}}{x_{23}}^{2}\\\
&&-2{x_{2}}{x_{23}}^{2}+4{x_{2}}+3{x_{23}}^{3}-4{x_{23}}^{2}-8{x_{23}}=0.\end{array}}\right\\}$
(26)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{2},x_{1},x_{23}]$ and an
ideal $I$ generated by $\\{g_{1},\,g_{2},\,g_{3},$
$\,z(\,x_{2}-x_{23})\,x_{1}\,x_{23}\,x_{2}-1\\}$ to find non-zero solutions of
equations (26) with $\,x_{2}\neq x_{23}$. We take a lexicographic order $>$
with $z>x_{2}>x_{1}>x_{23}$ for a monomial ordering on $R$. Then, by the aid
of computer, we see that a Gröbner basis for the ideal $I$ contains the
polynomials $\\{h(x_{23}),p_{1}(x_{23},x_{1}),p_{2}(x_{23},x_{2})\\}$, where
$h(x_{23})$ is a polynomial of $x_{23}$ given by

$\displaystyle
h(x_{23})=(n-6)^{2}(n-3)\left(n^{2}-7n+24\right){x_{23}}^{8}-2(n-6)^{2}(n-2)\left(n^{2}-n+6\right){x_{23}}^{7}$
$\displaystyle+(n-6)\left(n^{4}+26n^{3}-269n^{2}+686n-516\right){x_{23}}^{6}-44(n-6)(n-3)(n-2)(n+2){x_{23}}^{5}$
$\displaystyle+\left(14n^{4}+273n^{3}-3034n^{2}+5687n+1164\right){x_{23}}^{4}-2(n-2)\left(157n^{2}-157n-2778\right){x_{23}}^{3}$
$\displaystyle+\left(49n^{3}+1658n^{2}-6539n+836\right){x_{23}}^{2}-728(n-2)(n+5){x_{23}}+2704(n-1),$

$p_{1}(x_{23},x_{1})$ is a polynomial of $x_{23}$ and $x_{1}$ given by

$\displaystyle
p_{1}(x_{23},x_{1})=8(2n-5)\left(n^{2}-7n+27\right){x_{1}}+(n-6)^{3}(n-3)\left(n^{2}-7n+24\right){x_{23}}^{7}$
$\displaystyle-2(n-6)^{3}(n-2)\left(n^{2}-n+6\right){x_{23}}^{6}+(n-6)^{2}\left(n^{4}+19n^{3}-199n^{2}+371n-12\right){x_{23}}^{5}$
$\displaystyle-6(n-6)^{2}(n-2)\left(5n^{2}-5n-58\right){x_{23}}^{4}$
$\displaystyle+(n-6)\left(7n^{4}+140n^{3}-1641n^{2}+3090n+1248\right){x_{23}}^{3}-104(n-6)^{2}(n-2)(n+5){x_{23}}^{2}$
$\displaystyle+8\left(48n^{3}-625n^{2}+2305n-1719\right){x_{23}}$

and $p_{2}(x_{23},x_{2})$ is a polynomial of $x_{23}$ and $x_{2}$ given by

$\displaystyle
p_{2}(x_{23},x_{2})=-(n-6)^{2}(n-3)\left(n^{2}-7n+24\right)\left(2n^{2}-14n+15\right){x_{23}}^{7}$
$\displaystyle+2(n-6)^{2}(n-2)\left(n^{2}-n+6\right)\left(2n^{2}-14n+15\right){x_{23}}^{6}$
$\displaystyle-(n-6)\left(2n^{6}+25n^{5}-666n^{4}+3955n^{3}-8860n^{2}+7452n-2124\right){x_{23}}^{5}$
$\displaystyle+2(n-6)(n-2)\left(31n^{4}-248n^{3}+127n^{2}+2142n-2448\right){x_{23}}^{4}$
$\displaystyle-\left(15n^{6}+194n^{5}-5442n^{4}+33531n^{3}-73361n^{2}+38979n+18396\right){x_{23}}^{3}$
$\displaystyle+2(n-2)\left(119n^{4}-952n^{3}-1276n^{2}+21873n-28098\right){x_{23}}^{2}$
$\displaystyle-\left(7n^{5}+849n^{4}-11830n^{3}+53569n^{2}-79135n+24552\right){x_{23}}$
$\displaystyle+52(n-7)(n-1)\left(n^{2}-7n+27\right){x_{2}}+624(n-2)\left(n^{2}-7n+27\right).$

Thus we see that, if there exists a real root $x_{23}=\alpha_{23}$ of
$h(x_{23})=0$, then there are a real solution $x_{1}=\alpha_{1}$ of
$p_{1}(\alpha_{23},x_{1})=0$ and a real solution $x_{2}=\alpha_{2}$ of
$p_{2}(\alpha_{23},x_{2})=0$.

Now we have $h(0)=2704(n-1)>0$ for $n>1$,
$h(2)=4(16n^{5}-424n^{4}+4625n^{3}-25470n^{2}+70193n-77128)$ $=4(16(n-6)^{5}$
$+56(n-6)^{4}+209(n-6)^{3}+756(n-6)^{2}+1397(n-6)+1022)>0$ for $n\geq 6$ and
$h(1)=-2(n-9)(n-1)n^{2}<0$ for $n>9$. Note that for $n=9$
$h(6/5)=-1751152/390625<0$. Thus we see that the equation $h(x_{23})=0$ has
two positive roots $x_{23}=\alpha_{23},\beta_{23}$ with
$0<\alpha_{23}<1<\beta_{23}<2$ for $n>9$. For $n=9$ we have roots
$x_{23}=1,\beta_{23}$ with $6/5<\beta_{23}<2$.

Let $\gamma=\alpha_{23}$ or $\beta_{23}$. We have to show that the real
solutions $x_{1}=\alpha_{1}$ of $p_{1}(\gamma,x_{1})=0$ and $x_{2}=\alpha_{2}$
of $p_{2}(\gamma,x_{2})=0$ are positive. To this end, we take a lexicographic
order $>$ with $z>x_{2}>x_{23}>x_{1}$ for a monomial ordering on $R$. Then, by
the aid of computer, we see that a Gröbner basis for the ideal $I$ contains
the polynomial $h_{1}(x_{1})$ of $x_{1}$ given by

$\displaystyle
h_{1}(x_{1})=4(n-1)\left(n^{2}-7n+24\right)\left(n^{2}-7n+27\right)^{2}{x_{1}}^{8}-16(n-2)\left(n^{2}-7n+27\right)\times$
$\displaystyle(2n^{4}-28n^{3}+170n^{2}-504n+549){x_{1}}^{7}+(112n^{7}-2728n^{6}+29992n^{5}-192017n^{4}$
$\displaystyle+761574n^{3}-1849727n^{2}+2498826n-1434888){x_{1}}^{6}-4(n-2)(56n^{6}-1348n^{5}+13792n^{4}$
$\displaystyle-77805n^{3}+254449n^{2}-453225n+344070){x_{1}}^{5}+(280n^{7}-7880n^{6}+93373n^{5}-609014n^{4}$
$\displaystyle+2369548n^{3}-5476199n^{2}+6921202n-3695208){x_{1}}^{4}$
$\displaystyle-2(n-8)(n-2)\left(112n^{5}-2304n^{4}+18506n^{3}-73480n^{2}+144545n-109506\right){x_{1}}^{3}$
$\displaystyle+2(n-8)^{2}\left(56n^{5}-972n^{4}+6475n^{3}-20866n^{2}+32361n-18921\right){x_{1}}^{2}$
$\displaystyle-2(n-8)^{3}(n-2)(2n-5)\left(8n^{2}-64n+117\right){x_{1}}+(n-8)^{4}(n-3)(2n-5)^{2}.$

Now we have

$\displaystyle
h_{1}(x_{1})=4(n-1)\left(n^{2}-7n+24\right)\left(n^{2}-7n+27\right)^{2}{x_{1}}^{8}-16(n-2)\left(n^{2}-7n+27\right)\times$
$\displaystyle(2(n-8)^{4}+36(n-8)^{3}+266(n-8)^{2}+936(n-8)+1253){x_{1}}^{7}+(112(n-8)^{7}$
$\displaystyle+3544(n-8)^{6}+49576(n-8)^{5}+395823(n-8)^{4}+1933510(n-8)^{3}+5714577(n-8)^{2}$
$\displaystyle+9285018(n-8)+6127496){x_{1}}^{6}-4(n-2)(56(n-8)^{6}+1340(n-8)^{5}+13632(n-8)^{4}$
$\displaystyle+74259(n-8)^{3}+222137(n-8)^{2}+328423(n-8)+167678){x_{1}}^{5}+(280(n-8)^{7}$
$\displaystyle+7800(n-8)^{6}+91453(n-8)^{5}+578706(n-8)^{4}+2089420(n-8)^{3}+4129977(n-8)^{2}$
$\displaystyle+3804802(n-8)+1039208){x_{1}}^{4}-2(n-8)(n-2)(112(n-8)^{5}+2176(n-8)^{4}$
$\displaystyle+16458(n-8)^{3}+59368(n-8)^{2}+97185(n-8)+5203){x_{1}}^{3}+2(n-8)^{2}(56(n-8)^{5}$
$\displaystyle+1268(n-8)^{4}+11211(n-8)^{3}+48006(n-8)^{2}+97929(n-8)+73439){x_{1}}^{2}$
$\displaystyle-2(n-8)^{3}(n-2)(2n-5)(8(n-8)^{2}+64(n-8)+117){x_{1}}+(n-8)^{4}(n-3)(2n-5)^{2}.$

Thus we see that, for $n\geq 9$, the coefficients of the polynomial
$h_{1}(x_{1})$ are positive for even degree terms and negative for odd degree
terms, so if the equation $h_{1}(x_{1})=0$ has real solutions then these are
all positive.

We also take a lexicographic order $>$ with $z>x_{1}>x_{23}>x_{2}$ for a
monomial ordering on $R$. Then, by the aid of computer, we see that a Gröbner
basis for the ideal $I$ contains the polynomial $h_{2}(x_{2})$ of $x_{2}$
given by

$\displaystyle
h_{2}(x_{2})=64(n-6)^{2}(2n-5)^{2}\left(n^{2}-7n+27\right)^{2}{x_{2}}^{8}$
$\displaystyle-896(n-6)^{2}(n-2)(2n-5)\left(n^{2}-7n+27\right)\left(n^{2}-n+12\right){x_{2}}^{7}$
$\displaystyle+4(n-6)(1176n^{7}-14228n^{6}+100368n^{5}-730649n^{4}+4440678n^{3}$
$\displaystyle-18369941n^{2}+41390868n-33209244){x_{2}}^{6}-8(n-6)(n-2)(686n^{6}-2200n^{5}$
$\displaystyle+33593n^{4}-489642n^{3}+2433897n^{2}-7853838n+19276848){x_{2}}^{5}$
$\displaystyle+(2401n^{8}-2114n^{7}+85477n^{6}-3433940n^{5}+22264067n^{4}$
$\displaystyle-66085822n^{3}+304096111n^{2}-1233542964n+1558955520){x_{2}}^{4}$
$\displaystyle-2(n-2)(4949n^{6}-15874n^{5}+114730n^{4}-3099532n^{3}$
$\displaystyle+11930753n^{2}+23543310n-121637952){x_{2}}^{3}$
$\displaystyle+3\left(5099n^{6}-34656n^{5}+78010n^{4}-1041692n^{3}+8171395n^{2}-21025340n+15585312\right){x_{2}}^{2}$
$\displaystyle-104(n-2)\left(101n^{4}-661n^{3}+743n^{2}+5001n-15768\right){x_{2}}$
$\displaystyle+2704(n-3)(n-1)\left(n^{2}-7n+24\right)$

Now we have

$\displaystyle
h_{2}(x_{2})=64(n-6)^{2}(2n-5)^{2}\left(n^{2}-7n+27\right)^{2}{x_{2}}^{8}-896(n-6)^{2}(n-2)(2n-5)\times$
$\displaystyle\left(n^{2}-7n+27\right)\left(n^{2}-n+12\right){x_{2}}^{7}+4(n-6)(1176(n-8)^{7}+51628(n-8)^{6}+997968(n-8)^{5}$
$\displaystyle+10699111(n-8)^{4}+68192070(n-8)^{3}+256591483(n-8)^{2}+519880260(n-8)$
$\displaystyle+428454852){x_{2}}^{6}-8(n-6)(n-2)(686(n-8)^{6}+30728(n-8)^{5}+604153(n-8)^{4}$
$\displaystyle+6201974(n-8)^{3}+34466041(n-8)^{2}+95692802(n-8)+106856960){x_{2}}^{5}$
$\displaystyle+(2401(n-8)^{8}+151550(n-8)^{7}+4269685(n-8)^{6}+66669212(n-8)^{5}+617496227(n-8)^{4}$
$\displaystyle+3426718370(n-8)^{3}+11106086431(n-8)^{2}+19421585724(n-8)+14243093536){x_{2}}^{4}$
$\displaystyle-2(n-2)(4949(n-8)^{6}+221678(n-8)^{5}+4230810(n-8)^{4}+41090228(n-8)^{3}$
$\displaystyle+204389985(n-8)^{2}+502205726(n-8)+490441840){x_{2}}^{3}+3(5099(n-8)^{6}+210096(n-8)^{5}$
$\displaystyle+3586810(n-8)^{4}+31488548(n-8)^{3}+148970467(n-8)^{2}+362225908(n-8)$
$\displaystyle+357598976){x_{2}}^{2}-104(n-2)(101(n-8)^{4}+2571(n-8)^{3}+23663(n-8)^{2}+96825(n-8)$
$\displaystyle+147056){x_{2}}+2704(n-3)(n-1)\left(n^{2}-7n+24\right).$

Thus we see that, for $n\geq 8$, the coefficients of the polynomial
$h_{2}(x_{2})$ are positive for even degree terms and negative for odd degree
terms and that, so if the equation $h_{2}(x_{2})=0$ has real solutions then
these are all positive. Since the solutions satisfy the property $x_{2}\neq
x_{23}$, $x_{23}\neq 1$ and $x_{12}=x_{13}=1$, then Proposition 4.2 implies
that the metrics obtained are not naturally reductive ∎

From the above Propositions 6.1, 6.2 and 6.3 we obtain Theorem 1.1.

##  7\. The Lie groups $\operatorname{SO}((n-2)+1+1)$

In the present section we consider the scalar products (12) on the Lie groups
$\operatorname{SO}(k_{1}+k_{2}+k_{3})$, $k_{1}=n-2,k_{2}=k_{3}=1$, and prove
that for $n\geq 5$ we obtain only naturally reductive Einstein metrics. In
this case decomposition (8) becomes

$\mathfrak{so}((n-2)+1+1)=\mathfrak{m}_{1}\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23},$

where $\mathfrak{m}_{12}$, $\mathfrak{m}_{13}$ are equivalent as
$\operatorname{Ad}(\operatorname{SO}(n-2))$-modules.

######  Lemma 7.1.

For the metric (12) it is

$r(\mathfrak{m}_{12},\mathfrak{m}_{13})=(0).$

###### Proof.

We use the formula for the Ricci curvature in [Be, Corollary 7.38, p. 184]
adapted for the case of a Lie group. Then using polarization we obtain that

$r(X,Y)=-\frac{1}{2}\sum_{j}\langle[X,X_{j}],[Y,X_{j}]\rangle-\frac{1}{2}B(X,Y)+\frac{1}{4}\sum_{i,j}\langle[X_{i},X_{j}],X\rangle\langle[X_{i},X_{j}],Y\rangle,$
(27)

where $\\{X_{i}\\}$ is an orthonormal basis of $\mathfrak{so}(n)$ with respect
to the metric (12). For any $X\in\mathfrak{m}_{12},Y\in\mathfrak{m}_{13}$ we
need to show that $r(X,Y)=0$. A computation using (6) shows that
$\mathfrak{m}_{12},\mathfrak{m}_{13}$ are orthogonal with respect to $-B$, so
the second term in (27) vanishes.

We claim that (i) $\langle[X,X_{j}],[Y,X_{j}]\rangle=0$ and (ii)
$\langle[X_{i},X_{j}],X\rangle\langle[X_{i},X_{j}],Y\rangle=0$. Indeed, using
the orthonormal basis $\\{e_{ab}:1\leq a<b\leq n\\}$ of $\mathfrak{so}(n)$
introduced in Section 3 we let $X=e_{i,n-1}\ (1\leq i\leq n-2)$, $Y=e_{jn}\
(1\leq j\leq n-2)$.

To prove claim (i) let $X_{j}=e_{ab}\ (a<b)$. By using Lemma 3.1 it follows
that

$[X,X_{j}]=[e_{i,n-1},e_{ab}]=\begin{cases}\pm e_{ib},\quad\quad\ a=n-1\\\ \pm
e_{b,n-1},\quad a=i\\\ \pm e_{ia},\quad\quad\ b=n-1\\\ \pm e_{a,n-1},\quad
b=i\end{cases}$

and

$[Y,X_{j}]=[e_{jn},e_{ab}]=\begin{cases}\pm e_{jb},\quad a=n\\\ \pm
e_{bn},\quad a=j\\\ \pm e_{an},\quad b=j\\\ \pm e_{ja},\quad b=n.\end{cases}$

By using these expressions for the Lie brackets we can check that
$-B([X,X_{j}],[Y,X_{j}])=0$, hence for the metric (12) it is also
$\langle[X,X_{j}],[Y,X_{j}]\rangle=0$. To prove claim (ii) we let
$X_{i}=e_{ab}\ (a<b)$, $X_{j}=e_{cd}\ (c<d)$. By using Lemma 3.1 and obtain
that

$[X_{i},X_{j}]=[e_{ab},e_{cd}]=\begin{cases}\pm e_{ad},\quad b=c\\\ \pm
e_{ac},\quad b=d\\\ \pm e_{bd},\quad a=c\\\ \pm e_{bc},\quad a=d.\end{cases}$

The conclusion then follows by a similar argument as for the proof of claim
(i). ∎

######  Proposition 7.2.

For any $n\geq 5$, the only left-invariant Einstein metrics on the Lie group
$\operatorname{SO}(n)$, determined by the
$\operatorname{Ad}(\operatorname{SO}(n-2))$-invariant scalar products of the
form (12), are naturally reductive metrics.

###### Proof.

The proof involves manipulations of polynomials using Gröbner bases, and it is
quite extensive to be presented in its complete form. Since the metrics
obtained are only naturally reductive, we will only sketch the ideas behind
these computations.

We use the components for the Ricci tensor in Proposition 5.5 and consider the
system of equations

$r_{1}=r_{12},\,\,r_{12}=r_{23},\,\,r_{23}=r_{13}.$ (28)

Then finding Einstein metrics of the form (12) reduces to finding the positive
solutions of system (28), and we normalize our equations by putting
$x_{23}=1$. Then we have the system of equations:

$\left.{\begin{array}[]{l}g_{1}=n{x_{1}}^{2}{x_{13}}^{2}-2n{x_{1}}{x_{12}}{x_{13}}^{2}+n{x_{12}}^{2}{x_{13}}^{2}+{x_{1}}^{2}{x_{12}}^{2}-2{x_{1}}^{2}{x_{13}}^{2}-{x_{1}}{x_{12}}^{3}{x_{13}}\\\
\quad\quad+{x_{1}}{x_{12}}{x_{13}}^{3}+4{x_{1}}{x_{12}}{x_{13}}^{2}+{x_{1}}{x_{12}}{x_{13}}-4{x_{12}}^{2}{x_{13}}^{2}=0,\\\
g_{2}=-n{x_{1}}{x_{13}}+n{x_{12}}^{3}-2n{x_{12}}^{2}{x_{13}}+n{x_{12}}{x_{13}}^{2}+2n{x_{12}}{x_{13}}-n{x_{12}}+3{x_{1}}{x_{13}}\\\
\quad\quad-{x_{12}}^{3}+4{x_{12}}^{2}{x_{13}}-3{x_{12}}{x_{13}}^{2}-4{x_{12}}{x_{13}}+{x_{12}}=0,\\\
g_{3}=n{x_{1}}{x_{12}}-n{x_{12}}^{2}{x_{13}}+2n{x_{12}}{x_{13}}^{2}-2n{x_{12}}{x_{13}}-n{x_{13}}^{3}+n{x_{13}}-3{x_{1}}{x_{12}}\\\
\quad\quad+3{x_{12}}^{2}{x_{13}}-4{x_{12}}{x_{13}}^{2}+4{x_{12}}{x_{13}}+{x_{13}}^{3}-{x_{13}}=0.\end{array}}\right\\}$
(29)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{12},x_{13}]$ and an
ideal $I$ generated by $\\{g_{1},\,g_{2},$
$g_{3},\,z\,x_{1}\,x_{12}\,x_{13}-1\\}$ to find non zero solutions of
equations (29). We take a lexicographic order $>$ with $z>x_{13}>x_{12}>x_{1}$
for a monomial ordering on $R$. Then, by the aid of computer, we see that a
Gröbner basis for the ideal $I$ contains the polynomial

$({x_{1}}-1)\left((n-1){x_{1}}-(n-3)\right)\left(\left(n^{3}-2n^{2}+n-4\right){x_{1}}-(n-4)(n-1)^{2}\right)A(x_{1}),$
(30)

where

$\begin{array}[]{l}A(x_{1})=16(n-3)^{2}(n-2)^{3}(n-1){x_{1}}^{3}-4(n-3)(n-2)^{2}\left(5n^{3}-18n^{2}+21n-4\right){x_{1}}^{2}\\\
\qquad\qquad+4(n-2)^{2}(n-1)\left(n^{4}-27n^{2}+70n-32\right){x_{1}}\\\
\qquad\qquad-(n-4)(n-3)(n-1)^{2}n\left(n^{3}-6n^{2}+25n-32\right)\end{array}$

is a polynomial in $x_{1}$ of degree $3$. We divide our study in the following
cases:

Case (a) $A(x_{1})=0$. We claim that the system of equations (29) has no real
solutions in this case.

We consider the ideal $J$ generated by
$\\{g_{1},g_{2},g_{3},A(x_{1}),\,z\,x_{1}\,x_{12}\,x_{13}-1\\}$ of the
polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{12},x_{13}]$. We take a
lexicographic order $>$ with $z>x_{13}>x_{12}>x_{1}$ for a monomial ordering
on $R$. Then, by the aid of computer, we see that a Gröbner basis for the
ideal $J$ contains the polynomial

$\small{\begin{array}[]{l}p(x_{1},x_{12})=4(n-2)^{2}(n-1)\left(n^{4}-10n^{3}+37n^{2}-32n+16\right){x_{12}}^{2}-2(n-2)\times\\\
\left(\left(n^{4}+2n^{3}-3n^{2}-36n+64\right)(n-1)^{2}\right.+2(n-2)\left(2n^{5}-25n^{4}+93n^{3}-143n^{2}\right.\left.+93n-28\right){x_{1}}\\\
+\left.8(n-3)(n-2)^{2}(n+1)(n-1){x_{1}}^{2}\right){x_{12}}+(n-1)\left(4(n-3)(n-2)^{2}(n-1)\left(n^{2}-3n+4\right)\right.{x_{1}}^{2}\\\
-8(n-2)\left(3n^{4}-17n^{3}+31n^{2}-21n+8\right){x_{1}}\left.+(n-3)(n-1)\left(n^{3}-6n^{2}+25n-32\right)n^{2}\right).\end{array}}$

We view the above polynomial as a polynomial of the variable $x_{12}$, that is

$p(x_{1},x_{12})=\tilde{p}(x_{12})=4(n-2)^{2}(n-1)f_{2}(n){x_{12}}^{2}+f_{1}(x_{1},n)x_{12}+f_{0}(x_{1},n),$

where $f_{2}(n)=(n-4)^{4}+6(n-4)^{3}+13(n-4)^{2}+40(n-4)+96>0$ for $n\geq 4$.
We will show that, for the roots of the equation $A(x_{1})=0$, the polynomial
$\tilde{p}(x_{12})$ has no real roots for $x_{12}$. Indeed, the discriminant
$D_{\tilde{p}(x_{12})}$ of the polynomial $\tilde{p}(x_{12})$ has the form

$D_{\tilde{p}(x_{12})}=f_{1}^{2}-16(n-2)^{2}(n-1)f_{2}f_{0}=-4(n-2)^{2}\cdot\tilde{q}(x_{1}),$

where $\tilde{q}(x_{1})$ is a polynomial in $x_{1}$ of degree $4$. We need to
show that $D_{\tilde{p}(x_{12})}<0$ whenever $x_{1}=\alpha$, with
$A(\alpha)=0$ and $\alpha>0$ (since we are interested to Riemannian metrics).
By dividing the polynomial $\tilde{q}(x_{1})$ by $A(x_{1})$ we obtain that

$\tilde{q}(x_{1})=A(x_{1})B(x_{1})+q(x_{1}),$

where $q(x_{1})$ is a polynomial of degree $2$ given by

$q(x_{1})=4(n-1)f_{2}(n)(a_{0}(n)+a_{1}(n)x_{1}+a_{2}(n){x_{1}}^{2}).$

Then we need to show that $q(x_{1})>0$ or that

$r(x_{1})\equiv a_{0}(n)+a_{1}(n)x_{1}+a_{2}(n){x_{1}}^{2}>0,$

where the coefficients of the polynomial $r(x_{1})$ are explicitly given as

$\displaystyle a_{0}(n)$ $\displaystyle=$
$\displaystyle(n-1)(5n^{5}-38n^{4}+113n^{3}-196n^{2}+192n-64),$ $\displaystyle
a_{1}(n)$ $\displaystyle=$
$\displaystyle(n-2)(n^{6}-15n^{5}+69n^{4}-141n^{3}+150n^{2}-104n+32),$
$\displaystyle a_{2}(n)$ $\displaystyle=$ $\displaystyle
4(n-3)(n-2)^{2}(n-1)(n^{2}-3n+4)>0\ \ \mbox{for}\ n\geq 4.$

If $n\geq 9$ we expand the polynomials $a_{0}(n),a_{1}(n)$ as

$\displaystyle a_{0}(n)$ $\displaystyle=$ $\displaystyle
5(n-9)^{6}+227(n-9)^{5}+4291(n-9)^{4}+43197(n-9)^{3}$ $\displaystyle\
+244036(n-9)^{2}+732812(n-9)+912736,$ $\displaystyle a_{1}(n)$
$\displaystyle=$
$\displaystyle(n-9)^{7}+46(n-9)^{6}+882(n-9)^{5}+9036(n-9)^{4}+52353(n-9)^{3}$
$\displaystyle\ +164350(n-9)^{2}+229164(n-9)+48104,$

hence it follows that $r(x_{1})>0$.

If $5\leq n<9$ we consider the discriminant $D_{r(x_{1})}$ of the polynomial
$r(x_{1})$ and this is given by

$\displaystyle D_{r(x_{1})}$ $\displaystyle=$
$\displaystyle(n-2)^{3}(n^{11}-28n^{10}+227n^{9}-650n^{8}-761n^{7}+11240n^{6}-38635n^{5}$
$\displaystyle\ +75262n^{4}-93472n^{3}+72352n^{2}-31232n+5632).$

It is easy to see that for $n=5,6,7,8$ it is $D_{r(x_{1})}<0$ so $r(x_{1})>0$.
Therefore $D_{\tilde{p}(x_{12})}<0$ for all $n\geq 5$ and this completes the
proof of Case (a).

Case (b) $A(x_{1})\neq 0$. In this case we obtain only naturally reductive
Einstein metrics.

Indeed, from equation (30) we obtain the solutions

$x_{1}=1,\qquad x_{1}=\frac{n-3}{n-1},\qquad
x_{1}=\frac{(n-4)(n-1)^{2}}{n^{3}-2n^{2}+n-4}.$

By substituting the above solutions to the system (29) and computing Gröbner
bases for this system, we obtain the solutions

$(x_{1},x_{12},x_{13},x_{23})=(1,1,1,1),$
$(x_{1},x_{12},x_{13},x_{23})=\left(\frac{n-3}{n-1},1,\frac{n-3}{n-1},1\right),$
$(x_{1},x_{12},x_{13},x_{23})=\left(\frac{(n-4)(n-1)^{2}}{n^{3}-2n^{2}+n-4},\frac{(n-1)(n^{2}-3n+4)}{n^{3}-2n^{2}+n-4},\frac{(n-1)(n^{2}-3n+4)}{n^{3}-2n^{2}+n-4},1\right).$

From Proposition 4.4 it follows that the above metrics are naturally reductive
with respect to $\operatorname{SO}(n)\times L$, where $L$ is
$\operatorname{SO}(n)$, $\operatorname{SO}(n-1)$ or
$\operatorname{SO}(n-2)\times\operatorname{SO}(2)$ respectively. ∎

By working in a similar manner as in the above proofs, we can obtain Table 1
for the Lie groups
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$, which lists the
numbers of the new non naturally reductive and naturally reductive left-
invariant Einstein metrics of the forms (10), (11) or (12), up to isometry.
The table also contains results that due to space limitations we did not
provide explicit calculations.

$\operatorname{SO}(k_{1}+k_{2}+k_{3})$ | $(k_{1},k_{2},k_{3})$ | Non-naturally reductive | naturally reductive
---|---|---|---
$\operatorname{SO}(5)$ | $(3,1,1)$ | $0$ | $3$
| $(2,2,1)$ | $0$ | $3$
$\operatorname{SO}(6)$ | $(4,1,1)$ | $0$ | $3$
| $(3,2,1)$ | $0$ | $5$
| $(2,2,2)$ | $0$ | $2$
$\operatorname{SO}(7)$ | $(5,1,1)$ | $0$ | $3$
| $(4,2,1)$ | $0$ | $6$
| $(3,3,1)$ | ${\bf 1}$ | $5$
| $(3,2,2)$ | $0$ | $5$
$\operatorname{SO}(8)$ | $(6,1,1)$ | $0$ | $3$
| $(5,2,1)$ | $0$ | $6$
| $(4,3,1)$ | ${\bf 2}$ | $7$
| $(4,2,2)$ | $0$ | $5$
| $(3,3,2)$ | ${\bf 1}$ | $5$
$\operatorname{SO}(9)$ | $(7,1,1)$ | $0$ | $3$
| $(6,2,1)$ | $0$ | $6$
| $(5,3,1)$ | ${\bf 2}$ | $8$
| $(5,2,2)$ | $0$ | $5$
| $(4,3,2)$ | ${\bf 2}$ | $8$
| $(4,4,1)$ | ${\bf 2}$ | $5$
| $(3,3,3)$ | ${\bf 2}$ | $5$

Table 1. Numbers of new non naturally reductive and naturally reductive left-
invariant Einstein metrics on the Lie group
$\operatorname{SO}(n)=\operatorname{SO}(k_{1}+k_{2}+k_{3})$ up to isometry.
These are
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
metrics of the forms (10), (11) or (12).

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