# Classification of a Class of Solvable Real Lie Algebras by using Techniques
in Matrix Theory††thanks: Received by the editors on Month/Day/Year. Accepted
for publication on Month/Day/Year. Handling Editor: Namie of Handling Editor.
Corresponding Author: Vu Le Anh

Hieu Ha Van University of Economics and Law, Vietnam National University Ho
Chi Minh City, Vietnam<EMAIL_ADDRESS>
vula@uel.edu.vn). This research is funded by Vietnam National University Ho
Chi Minh City (VNU-HCM), under grant number B2022-34-05. Vu Le
Anh22footnotemark: 2 Hoa Duong Quang Hoa Sen University, Vietnam
(hoaduongquang@hoasen.edu.vn).

###### Abstract

We give a complete classification of the class of Lie algebras of simply
connected real Lie groups whose nontrivial coadjoint orbits are of codimension
1. Such a Lie group belongs to a well-known class, called the class of MD-
groups. The Lie algebra of an MD-group is called an MD-algebra. Some interest
properties of MD-algebras will be investigated as well. The techniques used in
this paper is elementary techniques in matrix theory and available to apply to
more general cases.

###### keywords:

Solvable Lie algebras, Lie groups, MD-algebras, MD-groups, Coadjoint orbits,
Classification of Lie groups, Commuting matrices, adjoint representation.

17B08,17B30.

##  1 Introduction

As we know, an $n$-dimensional Lie algebra can be defined via the commutators
of its basis vectors. More precisely, in order to define an $n$-dimensional
Lie structure $P$ on an $n$-dimensional vector space $\mathcal{G}$ spanned by
$\\{x_{1},x_{2},\dots,x_{n}\\}$, we need to define the Lie brackets

$[x_{i},x_{j}]=\sum\limits_{k=1}^{n}a_{ij}^{k}x_{k}\quad(1\leq i,j\leq n)$

which satisfy two properties:

(skew-symmetrix) $\displaystyle[x_{i},x_{j}]=-[x_{j},x_{i}]$
$\displaystyle\quad\forall i,j$ (Jacobi identity)
$\displaystyle\left[x_{i},[x_{j},x_{k}]\right]+\left[x_{j},[x_{k},x_{i}]\right]+\left[x_{k},[x_{i},x_{j}]\right]=0$
$\displaystyle\quad\forall i,j,k.$

In other words, we need to consider the family
$\\{\textnormal{ad}_{x_{i}}:i=1,2,..,n\\}$ of $n$ adjoint operators on
$\mathcal{G}$ which satisfies the following requirements:

(skew-symmetrix)
$\displaystyle\textnormal{ad}_{x_{i}}(x_{j})=-\textnormal{ad}_{x_{j}}(x_{i})$
$\displaystyle\quad\forall i,j$ (Jacobi identity)
$\displaystyle\textnormal{ad}_{x_{i}}\circ\textnormal{ad}_{x_{j}}-\textnormal{ad}_{x_{j}}\circ\textnormal{ad}_{x_{i}}=\textnormal{ad}_{[x_{i},x_{j}]}$
$\displaystyle\quad\forall i,j.$

Let’s denote by $A_{i}$ the $n\times n$ matrix of the adjoint operator
$\textnormal{ad}_{x_{i}}$ with respect to the basis
$\\{x_{1},\ldots,x_{n}\\}$, $i=1,\ldots,n$. Note that the family of matrices
$\mathcal{A}:=(A_{1},\ldots,A_{n})$ is determined by the $n$-dimensional Lie
structure $P$.

According to Kirillov et al. [13], the set $\mathcal{P}$ of all
$n$-dimensional Lie structures $P$ on $\mathcal{G}$ determines the set
$\mathcal{A}_{n}:=\\{\mathcal{A}:P\in\mathcal{P}\\}\subset\mathcal{G}\otimes\mathcal{G}^{*}\otimes\mathcal{G}^{*}$.
Due to the skew-symmetry and Jacobi identity, the set $\mathcal{A}_{n}$ forms
an algebraic veriety in $n^{3}$-dimensional space
$\mathcal{G}\otimes\mathcal{G}^{*}\otimes\mathcal{G}^{*}$ and it is called the
variety of $n$-dimensional Lie algebra structures in term of A. A. Kirillov et
al. [13].

Note that, by Kirillov et al. [13], the natural action of the general linear
group $GL(\mathcal{G})\equiv GL(n,\mathbb{R})$ on
$\mathcal{G}\otimes\mathcal{G}^{*}\otimes\mathcal{G}^{*}$ takes the variety
$\mathcal{A}_{n}$ into itself and the orbits of $GL(n,\mathbb{R})$ in
$\mathcal{A}_{n}$ correspond to the isomorphic classes of $n$-dimensional Lie
algebras. In other words, the orbit space $\mathcal{A}_{n}/GL(n,\mathbb{R})$
coincides with the set of isomorphic classes of $n$-dimensional real Lie
algebra. Hence, the number of parameters increases drastically and the volume
of calculations, therefore, will become enormous when $n$ not small [16]. That
explains why the problem of classification of Lie algebras (as well as Lie
groups) of dimension $n$ strictly greater than 6 is still open, in general,
until now. In some certain cases, the problem of the classification is proven
to be “will” and “hopeless” [5, 6, 7, 8].

However, it is still possible to classify some interest subclass of solvable
Lie algebras (of general finite dimensions) added one or some special
properties in which we can use techniques in matrix theory to make the
matrices in the family $\mathcal{A}=(A_{1},\ldots,A_{n})$ to be as simple as
possible. This paper will investigate such a class, called MD-algebras.
Basically, we know that Lie groups whose coadjoint orbits are all trivial
(i.e. of dimension 0) are trivial. But, in the case they have non-trivial
coadjoint orbits, the problem of classification is not easy and still open.
The basis idea of MD-groups (as well as MD-algebras) is to consider Lie groups
(and their Lie algebras) in which all non-trivial coadjoint orbits are of the
same dimension. This idea comes from the Kirillov’s method [12] and propsed by
Do Ngoc Diep while searching for the class of Lie groups whose $C^{*}$-algebra
can be characterized by BDF K-functions in 1980 [4]. More precisely, Diep
recommended to consider finite-dimensional solvable and simply connected Lie
groups whose $K$-orbits are either of dimension 0 or of a constant positive
dimension (which is called the maximal dimension). This Lie group is said to
have the property MD and is called an MD-group. The Lie algebra of an MD-group
is called a Lie algebra having the property MD or an MD-algebra for brevity.
It is worth pointing out that the family of $K$-orbits of the maximal
dimension of an MD-group forms a measured foliation in terms of A. Connes [14,
3].

The problem of classifying MD-algebras has received much attention in recent
decades, but the results are not much.

  * •

In 1984, Vuong Manh Son and Ho Huu Viet gave a complete classification of MD-
groups whose maximal dimension of $K$-orbits is equal to the dimension of the
group [20].

  * •

In 1990, Vu L. A. (the second author of this paper) gave the classification
(up to isomorphism) of all MD-algebras of dimension 4 [14, 15].

  * •

In 1995, D. Arnal et al. gave a list of all MD-algebras in which the maximal
dimension of $K$-orbits of corresponding MD-groups is 2 [1].

  * •

In 2012, Vu L. A., Hieu H. V. (the first author of this paper) et al. gave the
classification (up to isomorphism) of MD-algebras of dimension 5 [22, 17].

  * •

In 2016, Vu L. A., Hieu H. V. et al. classified (up to isomorphism) all of MD-
algebras which have the first derived ideal of codimension 1 or 2 [21].

  * •

In 2019, M. Goze et al. gave the classification (up to isomorphism) of all of
MD-algebras such that their nontrivial $K$-orbits are of dimension 4 [10].

Basically, the approaches of the classifying MD-algebras obtained in the above
papers can be divided into the following directions: (1) fixing the dimensions
of Lie algebras and (or) the dimensions of the derived series [14, 15, 22, 17,
21], (2) fixing the maximal dimension of K-orbits [1, 10], and (3) fixing the
codimension of K-orbits [20]. Our paper follows the third direction.
Specifically, we will classify all MD-algebras whose nontrivial $K$-orbits are
of codimension 1 (i.e. the maximal dimension of $K$-orbits is 1 less than the
dimension of such a Lie algebra).

The paper is organized into three sections, including this introduction. In
Section 2 we recall some basic preliminary concepts, notations and properties
which will be used throughout the paper. The last section will be devoted to
setting and proving the main results of the paper.

##  2 Preliminaries

Throughout this paper, the following notations will be used.

  * •

An $n\times n$ matrix whose $(i,j)$-entry is $a_{ij}$ will be written as
$(a_{ij})_{n\times n}$. While the $(i,j)$-entry of a matrix $A$ will be
denoted by $(A)_{ij}$. To simplify notation we use the same letter $A$ for the
endomorphism of $\mathbb{F}^{n}$ which is defined by assigning a column vector
$v\in\mathbb{F}^{n}$ to $Av$, assumed that the entries of $A$ take values in
$\mathbb{F}$. Note that $\operatorname*{Im}A$ is equal to the
$\mathbb{F}$-vector space spanned by the columns of $A$. We shall denote by
$\boldsymbol{0}$ the zero matrix of suitable size.

  * •

With the notation $V=\langle v_{1},v_{2},\dots,v_{n}\rangle$, we mean
$v_{1},v_{2},\dots,v_{n}$ is a basis of the vector space $V$. The dual space
of $V$ will be denoted by $V^{*}$. It is well-know that if $V=\langle
v_{1},\dots,v_{n}\rangle$ then $V^{*}=\langle
v_{1}^{*},\dots,v_{n}^{*}\rangle$ where each $v_{i}^{*}$ is defined by
$v_{i}^{*}(v_{j})=\delta_{ij}$ (the Kronecker delta symbol) for $1\leq i,j\leq
n$.

  * •

By $\mathcal{G}$ we always mean a solvable Lie algebra of finite dimension
over the real field with Lie bracket $[\cdot,\cdot]$. For any
$x\in\mathcal{G}$, we will denote by $\textnormal{ad}_{x}$ the adjoint action
of $x$ on $\mathcal{G}$, i.e. $\textnormal{ad}_{x}$ is the endomorphism of
$\mathcal{G}$ defined by $\textnormal{ad}_{x}(y)=[x,y]$ for every
$y\in\mathcal{G}$. By $\textnormal{ad}^{1}_{x}$, $\textnormal{ad}^{2}_{x}$ we
mean the restricted maps of $\textnormal{ad}_{x}$ on the first derived ideal
$\mathcal{G}^{1}:=[\mathcal{G},\mathcal{G}]$ and the second derived ideal
$\mathcal{G}^{2}:=[\mathcal{G}^{1},\mathcal{G}^{1}]$, respectively. Since both
$\mathcal{G}^{1}$ and $\mathcal{G}^{2}$ are ideals of $\mathcal{G}$, we shall
consider $\textnormal{ad}^{1}_{x}$ and $\textnormal{ad}^{2}_{x}$ as
endomorphisms of $\mathcal{G}^{1}$ and $\mathcal{G}^{2}$, respectively.
Similary, we will denote by $F^{1}$ the restriction $F|_{\mathcal{G}^{1}}$ of
any form $F\in\mathcal{G}^{*}$ on $\mathcal{G}^{1}$.

Next, we will recall some definitions, elementary results about coadjoint
representation of solvable Lie groups. For more details, we refer reader to
[12].

######  Definition 2.1.

Let $G$ be a Lie group and let $\mathcal{G}$ be its Lie algebra. If
$\text{Ad}:G\rightarrow\text{Aut}(\mathcal{G})$ denotes the adjoint
representation of $G$. Then the action

$\begin{array}[]{lrll}K:&G&\rightarrow&\text{Aut}(\mathcal{G}^{*})\\\
&g&\mapsto&K_{g}\end{array}$

defined by

$K_{g}(F)(x)=F\bigl{(}\text{Ad}(g^{-1})(x)\bigr{)};\text{ for
}F\in\mathcal{G}^{*},x\in\mathcal{G}$

is called the coadjoint representation of $G$ in $\mathcal{G}^{*}$. Each orbit
in the coadjoint representation of $G$ is called a coadjoint orbit (or a
$K$-orbit for short) of $G$ as well as of $\mathcal{G}$.

For each $F\in\mathcal{G}^{*}$, the $K$-orbit that passes through the point
$F$ is denoted by $\Omega_{F}$, i.e.

$\Omega_{F}=\\{K_{g}(F):g\in G\\}.$

######  Definition 2.2.

[12, Section 15.1] Let $F$ be any element in $\mathcal{G}^{*}$. The Kirillov’s
bilinear form with respect to $F$ is defined by

$B_{F}(x,y):=F([x,y]);\text{ for }x,y\in\mathcal{G}.$

It is a bilinear skew symmetric form on $\mathcal{G}$.

The matrix of $B_{F}$ with respect to a basis $\\{x_{1},x_{2},\dots,x_{n}\\}$
of $\mathcal{G}$, which is equal to $\bigl{(}F([x_{i},x_{j}])\bigr{)}_{n\times
n}$, is called the Kirillov’s matrix of $F$ with respect to that basis. From
now on, when a basis of $\mathcal{G}$ is fixed, we will always treat $B_{F}$
as the Kirillov’s matrix of $F$ with respect to the fixed matrix. The set of
matrices $\\{B_{F}:F\in\mathcal{G}^{*}\\}$ will be called the Kirillov’s
matrices of $\mathcal{G}$. The following proposition gives us a connection
between dimension of $\Omega_{F}$ and the rank of $B_{F}$.

######  Proposition 2.3.

[12, Section 15.3] Let $F$ be any element in $\mathcal{G}^{*}$. Then

$\dim\Omega_{F}=\operatorname*{rank}{B_{F}}.$

As a consequence, $\dim\Omega_{F}$ is always even for every $F$ in
$\mathcal{G}^{*}$ and $\dim\Omega_{F}>0$ if and only if $F^{1}$ is not equal
to zero, i.e. $F|_{\mathcal{G}^{1}}\neq 0$. It is trivial to see that if
$\dim\Omega_{F}=0$ for every $F\in\mathcal{G}^{*}$ then $\mathcal{G}$ is
commutative. Our concern will be Lie algebras in which the dimensions of non-
trivial $K$-orbits are equal.

######  Definition 2.4.

[20, Introduction] A finite dimensional solvable and simply connected Lie
group is called an MD-group (in terms of Do Ngoc Diep) if its $K$-orbits are
either of dimension zero (trivial ones) or of the same positive constant
dimension. The Lie algebra of an MD-group is called an MD-algebra.

Let $G$ be an MD-group. Then the positive constant dimension of $K$-orbits is
also called the maximal dimension of $K$-orbits of $G$ (as well as of the
corresponding Lie algebra $\mathcal{G}$). If this maximal dimension is given
by $k$ ($k>0$), we also say that $\mathcal{G}$ is an MDk-algebra. Furthermore,
if $\dim\mathcal{G}=n$ then $\mathcal{G}$ is also called an
MD${}_{k}(n)$-algebra. In particular, if an MD-group $G$ satisfies one of the
following properties: either all $K$-orbits are trivial or the maximal
dimension of $K$-orbits is equal to the dimension of $G$ then $G$ is said to
be an SMD-group. The Lie algebra of an SMD-group is called an SMD-algebra. It
is clear that any commutative Lie algebra is an SMD-algebra and it has no
maximal dimension of $K$-orbits because its $K$-orbits are all trivial.

In order to classify MD${}_{n-1}(n)$-algebras, we firstly note that all
solvable Lie algebras of dimension $n<4$ are obviously MD-algebras and can be
classified easily. Therefore we only take interest in MD-algebras of dimension
$n\geq 4$. On the other hand, in 1984 Vuong Manh Son and Ho Huu Viet gave a
bound for the solvable index of MD-algebras which is presented in the
following proposition.

######  Proposition 2.5.

[20, Theorem 4] If $\mathcal{G}$ is an MD-algebra then its second derived
ideal $\mathcal{G}^{2}$ is commutative.

Therefore, if $\mathcal{G}$ is an MD-algebra then the third derived ideal
$\mathcal{G}^{3}:=[\mathcal{G}^{2},\mathcal{G}^{2}]$ is the trivial vector
space. In this paper, a solvable Lie algebra is said to be $i$-step solvable
if $\mathcal{G}^{i}$ is non trivial (i.e. $\mathcal{G}^{i}\neq\\{0\\}$) and
commutative. By using this term, Proposition 2.5 means that any non-
commutative MD-algebra is either 1-step or 2-step solvable. The basis idea of
the classification we use in this paper is to classify 1-step solvable
MD${}_{n-1}(n)$-algebras firstly and to connect 2-step solvable
MD${}_{n-1}(n)$-algebras with 1-step solvable MD${}_{n-1}(n-1)$-algebras
afterwards. To deal with 1-step solvable ones, it is necessary to recall the
following classical result.

######  Proposition 2.6.

[22, Lemma 3.1] If $\mathcal{G}$ is 1-step solvable then

$\textnormal{ad}^{1}_{x}\circ\textnormal{ad}^{1}_{y}=\textnormal{ad}^{1}_{y}\circ\textnormal{ad}^{1}_{x}\quad\forall
x,y\in\mathcal{G}.$

In other words, $\\{\textnormal{ad}^{1}_{x}:x\in\mathcal{G}\\}$ is a commuting
family of linear endomorphisms of $\mathcal{G}^{1}$.

Hence, it is reasonable to consider the set of commuting matrices over the
real field. It is well-known that an arbitrary set of commuting matrices over
the complex field $\mathbb{C}$ has an invariant subspace of dimension 1.
Therefore, it may be simultaneously brought to triangular form by a unitary
similarity [18, 11]. Similarly for the triangular form of an arbitrary set of
commuting real matrices. We state these classical results in the following
Proposition.

######  Proposition 2.7.

Let $\mathcal{S}\subseteq M_{n\times n}(\mathbb{F})$ be a set of commuting
$n\times n$ matrices over a field $\mathbb{F}$, i.e. $AB=BA$ for every
$A,B\in\mathcal{S}$.

  * (i)

If $\mathbb{F}=\mathbb{C}$ then $\mathcal{S}$ is simultaneously
triangularizable by a unitary matrix, i.e there exists a complex unitary
matrix $T$ so that $T^{-1}AT$ is upper triangular matrix for all
$A\in\mathcal{S}$.

  * (ii)

if $\mathbb{F}=\mathbb{R}$ then $\mathcal{S}$ is block simultaneously
triangularizable by a orthogonal matrix, i.e. there is a real orthogonal
matrix $T$ so that $T^{-1}AT$ is a block upper triangular matrix, where each
diagonal block is of size either 1 or 2, for all $A\in\mathcal{S}$.

##  3 Main Results

In this section, we firstly give some new interest properties of
MD${}_{k}(n)$-algebras, especially 1-step solvable MD-algebras [Theorem 3.1,
3.3 & 3.7]. Secondly, we give a connection between 1-step solvable MD-algebras
and 2-step solvable MD-algebras [Theorem 3.9]. Finally, we will present the
complete classification of $\text{MD}_{n-1}(n)$-algebras in the last theorem.
It is possible that we can apply Theorem 3.3 and Theorem 3.9 to obtain the
classification of MD${}_{k}(n)$-algebras with $n-k$ small (not necessary to be
1) but we will not develop this point here.

###  3.1 Some new characteristics of MD-algebras

######  Theorem 3.1.

Let $\mathcal{G}$ be a non-commutative Lie algebra. Then $\mathcal{G}$ is a
decomposable MDk-algebra if and only if it is a trivial extension of an
indecomposable MDk-algebra by a finite-dimensional commutative Lie algebra.

######  Proof 3.2.

Let’s first prove the “if” part. Assume that $\mathcal{G}$ is a direct sum of
an MDk-algebra with a finite-dimensional commutative algebra $\mathbb{R}^{r}$,
i.e. $\mathcal{G}=\mathcal{K}\oplus\mathbb{R}^{r}$, where $\mathcal{K}$ is an
MDk-algebra. If so, we have

(3.1) $\mathcal{G}^{1}=\mathcal{K}^{1}.$

Denote by $n$ the dimension of $\mathcal{G}$. Let’s fix a basis
$\\{x_{1},x_{2},\dots,x_{n}\\}$ of $\mathcal{G}$ so that $\mathcal{K}=\langle
x_{1},\dots,x_{n-r}\rangle$. For any
$F=F_{\mathcal{K}}+F_{r}\in\mathcal{G}^{*}$ with
$F_{\mathcal{K}}\in\mathcal{K}^{*}\subseteq\mathcal{G}^{*}$ and
$F_{r}\in(\mathbb{R}^{r})^{*}\subseteq\mathcal{G}^{*}$, we denote by
${\bar{\Omega}}_{F_{\mathcal{K}}}$ the $K$-orbit of $F_{\mathcal{K}}$ in
$\mathcal{K}^{*}\subset\mathcal{G}^{*}$. It is elementary to see that

(3.2) $B_{F}=\begin{bmatrix}{\bar{B}}_{F_{\mathcal{K}}}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}.$

where ${\bar{B}}_{F_{\mathcal{K}}}$ is the Kirillov’s matrix of
$F_{\mathcal{K}}$ in $\mathcal{K}$ with respect to the basis
$\\{x_{1},\dots,x_{n-r}\\}$. Therefore, we get
$\operatorname*{rank}B_{F}=\operatorname*{rank}{\bar{B}}_{F_{\mathcal{K}}}$.
Since $\mathcal{K}$ is an MDk-algebra, we have

(3.3)
$\dim{\Omega_{F}}=\dim{\bar{\Omega}}_{F_{\mathcal{K}}}=\left\\{\begin{array}[]{ll}0&\text{if
}F|_{\mathcal{K}^{1}}=\boldsymbol{0}\\\ k&\text{elsewhere}\end{array}\right.$

This proves $\mathcal{G}$ is an MDk-algebra, as required.

It remains to prove the “only if” part. By contradiction, assume that
$\mathcal{G}=\mathcal{K}\oplus\mathcal{L}$ in which both $\mathcal{K}$ and
$\mathcal{L}$ are non-commutative sub-algebras of $\mathcal{G}$. Let’s denote
by $n$ and $m$ the dimensions of $\mathcal{G}$ and $\mathcal{K}$,
respectively. Since $\mathcal{G}=\mathcal{K}\oplus\mathcal{L}$, there is a
basis $\\{x_{1},x_{2},\dots,x_{n}\\}$ of $\mathcal{G}$ so that

(3.4) $\left\\{\begin{array}[]{l}\mathcal{K}=\langle
x_{1},x_{2},\dots,x_{m}\rangle,\\\ \mathcal{L}=\langle
x_{m+1},\dots,x_{n}\rangle.\end{array}\right.$

Because both $\mathcal{K}$ and $\mathcal{L}$ are non-commutative, there exist
two non-zero elements $F_{1},F_{2}\in(\mathcal{G})^{*}$ so that

(3.5)
$\left\\{\begin{array}[]{ll}F_{1}|_{\mathcal{K}^{1}}\neq\boldsymbol{0},&F_{1}|_{\mathcal{L}^{1}}=\boldsymbol{0},\\\
F_{2}|_{\mathcal{K}^{1}}=\boldsymbol{0},&F_{2}|_{\mathcal{L}^{1}}\neq\boldsymbol{0}.\end{array}\right.$

If so, it is elementary to see that

(3.6) $B_{F_{1}}=\left[\begin{array}[]{cl}M&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right],\text{ and
}B_{F_{2}}=\left[\begin{array}[]{cl}\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&N\end{array}\right],$

where $M$ and $N$ are two non-zero matrices of dimension $m\times m$ and
$(n-m)\times(n-m)$, respectively. We thus get

$0<\operatorname*{rank}{B_{F_{1}}}<\operatorname*{rank}{B_{F_{1}}}+\operatorname*{rank}{B_{F_{2}}}=\operatorname*{rank}{B_{F_{1}+F_{2}}}.$

In other words,

$0<\dim\Omega_{F_{1}}<\dim\Omega_{F_{1}+F_{2}},$

a contradiction to the fact that $\mathcal{G}$ is an MD-algebra. This
completes the proof.

######  Theorem 3.3.

Let $\mathcal{G}$ be a 1-step solvable $\text{MD}_{k}(n)$-algebra so that
$\dim\mathcal{G}^{1}>n-k$. Then we have the following assertions.

  1. 1.

$\sum\limits_{y\in\mathcal{G}}\operatorname*{Im}\textnormal{ad}^{1}_{y}=\mathcal{G}^{1}.$

  2. 2.

There is $x$ in $\mathcal{G}$ so that $\textnormal{ad}^{1}_{x}$ is an
automorphism on the vector space $\mathcal{G}^{1}$.

  3. 3.

$\mathcal{G}$ can be expressed as the semidirect sum
$\mathcal{L}\oplus_{\rho}\mathcal{G}^{1}$ of a commutative subalgebra
$\mathcal{L}$ of $\mathcal{G}$ with the first derived ideal $\mathcal{G}^{1}$
by the representation $\rho:\mathcal{L}\rightarrow Der(\mathcal{G}^{1})$,
which is defined by $\rho(x)=\textnormal{ad}^{1}_{x}$ for every
$x\in\mathcal{L}$.

In order to prove this Theorem, we will need the following lemma.

######  Lemma 3.4.

Let $\mathcal{S}=\\{A_{1},A_{2},\dots,A_{s}\\}$ be a set of complex upper
triangular matrices of size $m\times m$. If $\mathcal{S}$ is commutative (i.e.
$AB=BA$ for every $A,B\in\mathcal{S}$) and

(3.7) $\sum\limits_{i=1}^{s}\operatorname*{Im}A_{i}=\mathbb{C}^{m}$

then there is a linear combination of elements of $\mathcal{S}$ so that it is
nonsingular.

######  Proof 3.5.

The proof is by induction on $m$. It is trivial for $m=1$. Since $A_{i}$ is of
upper triangular form for every $i$, the equality (3.7) implies that at least
one of $\\{(A_{i})_{mm}:i=1,\dots,s\\}$ is non-zero (if not, the vector
$(0,0,\dots,1)^{t}\notin\sum\limits_{i=1}^{s}\operatorname*{Im}A_{i}=\mathbb{C}^{m}$,
a contradiction). By re-indexing if necessary, we may assume that
$(A_{1})_{mm}\neq 0$. Furthermore, the proof is still correct if we replace
$\mathcal{S}$ by
$\mathcal{S}^{\prime}=\\{A_{1},A_{2}-\beta_{2}A_{1},\dots,A_{s}-\beta_{s}A_{1}\\}$
for every $\beta_{2},\dots,\beta_{s}\in\mathbb{C}$. Therefore, we may assume
from beginning that

(3.8) $\left\\{\begin{array}[]{ll}(A_{i})_{mm}=0&\forall i\geq 2,\\\
(A_{1})_{mm}\neq 0.\end{array}\right.$

Since $A_{1}A_{i}=A_{i}A_{1}$ for every $i$, we have

(3.9) $(A_{1}A_{i})_{jm}=(A_{i}A_{1})_{jm}\text{ for every }i,j.$

Therefore, for every $i$, the last column of $A_{i}$ is a linear combination
of the following columns: all the columns of $A_{1}$ and the first $(m-1)$
columns of $A_{i}$. It follows that
$\sum\limits_{i=1}^{s}\operatorname*{Im}A_{i}$ is spanned by the last column
of $A_{1}$ and the first $m-1$ columns of $A_{1},\dots,A_{s}$. We conclude
from the equality (3.7) that $\mathbb{C}^{m-1}$ is spanned by the first
$(m-1)$ columns of $A_{1},\dots,A_{s}$.

In other words, if we denote by $B_{i}$ the $(m-1)\times(m-1)$ matrix obtained
by deleting the $m$-th column and $m$-th row of $A_{i}$ for each $1\leq i\leq
s$ then $\\{B_{1},B_{2},\dots,B_{s}\\}$ satisfies the following properties:

  * •

it is a set of commuting (complex) matrices of dimension $(m-1)\times(m-1)$,

  * •

each $B_{i}$ is of upper triangular form for every $i$, and

  * •

$\mathbb{C}^{m-1}$ is spanned by the columns of $B_{1},B_{2},\dots,B_{s}$,
i.e.

(3.10) $\sum\limits_{i=1}^{s}\operatorname*{Im}B_{i}=\mathbb{C}^{m-1}.$

By induction, there are $\alpha_{1},\alpha_{2},\dots,\alpha_{s}\in\mathbb{C}$
so that $\alpha_{1}B_{1}+\cdots+\alpha_{s}B_{s}$ is nonsingular. Remark that
we can always choose $\alpha_{1}\neq 0$ (if $\alpha_{1}=0$ then
$\alpha_{2}B_{2}+\cdots+\alpha_{s}B_{s}$ is nonsingular and hence, there
exists $\alpha_{1}^{\prime}\neq 0$ so that
$\alpha^{\prime}_{1}B_{1}+\alpha_{2}B_{2}+\cdots+\alpha_{s}B_{s}$ is
nonsingular because the equation
$\det\bigl{(}B_{1}+x(\alpha_{2}B_{2}+\cdots+\alpha_{s}B_{s})\bigr{)}=0$ has
finite solutions). With the assumption $\alpha_{1}\neq 0$ in hand, it follows
immediately from the equation (3.8) and from the nonsingularity of
$\alpha_{1}B_{1}+\cdots+\alpha_{s}B_{s}$ that

$\alpha_{1}A_{1}+\cdots+\alpha_{s}A_{s}$

is nonsingular, which proves the lemma.

######  Proof 3.6 (Proof of Theorem 3.3).

Set $m=\dim\mathcal{G}^{1}$. Throughout this proof, we fix a basis
$\\{x_{1},x_{2},\dots,x_{n}\\}$ of $\mathcal{G}$ so that
$\mathcal{G}^{1}=\langle x_{n-m+1},x_{n-m+2},\dots,x_{n}\rangle$.

  1. 1.

By contradiction, assume that
$\sum\limits_{y\in\mathcal{G}}\operatorname*{Im}\textnormal{ad}^{1}_{y}\subsetneq\mathcal{G}^{1}$.
Then there is an element $F\in(\mathcal{G})^{*}$ so that

(3.11) $\left\\{\begin{array}[]{ll}F^{1}\neq 0\\\
F|_{\sum\limits_{y\in\mathcal{G}}\operatorname*{Im}\textnormal{ad}^{1}_{y}}=0&\end{array}\right.$

If so, it is clear that

(3.12) $F([x_{i},x_{j}])=0,\text{ unless }1\leq i,j\leq n-m.$

It follows from Proposition 2.3 that $k=\dim\Omega_{F}\leq n-m<k$, a
contradiction. This proves the first item.

  2. 2.

Let $\mathcal{G}_{\mathbb{C}}=\langle x_{1},\dots,x_{n}\rangle_{\mathbb{C}}$
be the complexification of $\mathcal{G}$. By Proposition 2.6,
$\\{\textnormal{ad}^{1}_{x_{i}}:i=1,..,n-m\\}$ is a family of commuting
endomorphisms of the real type. Hence, there is a basis of
$\mathcal{G}_{\mathbb{C}}^{1}$ so that the matrix of
$\textnormal{ad}_{x_{i}}|_{\mathcal{G}_{\mathbb{C}}^{1}}$ with respect to that
basis is of upper triangular form for every $1\leq i\leq n-m$ (by Proposition
2.7). It follows from the first item and Lemma 3.4 that there is a complex
linear combination $x$ of $x_{1},x_{2},\dots,x_{n-m}$ so that
$\textnormal{ad}_{x}|_{\mathcal{G}_{\mathbb{C}}^{1}}$ is isomorphic.
Therefore, by using the same notation
$\textnormal{ad}^{1}_{x_{i}}(i=1,..,n-m)$ for their matrices with respect to
that the basis $\\{x_{1},\dots,x_{n}\\}$, we easily see that the polynomial

$\det(\zeta_{1}\textnormal{ad}^{1}_{x_{1}}+\cdots+\zeta_{n-m}\textnormal{ad}^{1}_{x_{n-m}})$

of variables $\zeta_{1},\dots,\zeta_{n-m}$ does not vanish. Hence, there is
$\alpha_{1},\dots,\alpha_{n-m}\in\mathbb{R}$ so that

(3.13)
$\det(\alpha_{1}\textnormal{ad}^{1}_{x_{1}}+\cdots+\alpha_{n-m}\textnormal{ad}^{1}_{x_{n-m}})\neq
0.$

This prove the existence of a real linear combination $x^{\prime}$ of
$x_{1},x_{2},\dots,x_{n-m}$ so that $\textnormal{ad}^{1}_{x^{\prime}}$ is
nonsingular, as required.

  3. 3.

It follows from the second item proven above that we may assume (after
changing basis if necessary) $\textnormal{ad}^{1}_{x_{1}}$ is nonsingular.
Hence, for each $j\geq 2$, there are
$\alpha_{j1},\alpha_{j2},\dots,\alpha_{jm}\in\mathbb{R}$ so that

(3.14)
$[x_{1},x_{j}]+\alpha_{j1}[x_{1},x_{n-m+1}]+\cdots+\alpha_{jm}[x_{1},x_{n}]=0.$

Therefore, by basis changing
$x^{\prime}_{j}=x_{j}+\sum\limits_{i=1}^{m}\alpha_{ji}x_{n-m+i}$ if necessary,
we can assume from beginning that $[x_{1},x_{j}]=0$ for all $1\leq j\leq n-m$.
With this assumption in hand, we may apply the Jacobi’s identity for
$(x_{1},x_{j},x_{k})$ to obtain

(3.15) $[x_{1},[x_{j},x_{k}]]=0\text{ for all }1\leq j,k\leq n-m.$

It follows from the nonsingularity of $\textnormal{ad}^{1}_{x_{1}}$ that
$[x_{j},x_{k}]=0$ for all $1\leq j,k\leq n-m.$ Hence, $\mathcal{G}$ is
isomorphic to the semidirect sum of $\mathcal{G}^{1}$ with the commutative
subalgebra $\mathcal{L}:=\langle
x_{1},\dots,x_{n-m}\rangle\cong{\mathbb{R}}^{n-m}$ of $\mathcal{G}$, as
required.

Theorem 3.3 gains in interest if we realize that, by fixing a basis
$\\{x_{1},x_{2},\dots,x_{n}\\}$ of the Lie algebra
$\mathcal{L}\oplus_{\rho}\mathcal{G}^{1}$ so that $\mathcal{L}=\langle
x_{1},\dots,x_{n-m}\rangle$ and $\mathcal{G}^{1}=\langle
x_{n-m+1},\dots,x_{n}\rangle$, the Kirilov’s matrix of any
$F\in(\mathcal{L}\oplus_{\rho}\mathcal{G}^{1})^{*}$ is of the following form

$\begin{bmatrix}\boldsymbol{0}&M_{F}\\\
-(M_{F})^{t}&\boldsymbol{0}\end{bmatrix}$

where $M_{F}$ is some matrix of size $(n-m)\times m$. It turns out that the
dimension of each $\Omega_{F}$ depends on the rank of $M_{F}$. More precisely,
$\dim\Omega_{F}=2\operatorname*{rank}{M_{F}}$.

Besides, it is easily seen that the $i$-th row of $M_{F}$ is exactly

$\bigl{(}F^{1}\circ\textnormal{ad}^{1}_{x_{i}}(x_{n-m+1}),F^{1}\circ\textnormal{ad}^{1}_{x_{i}}(x_{n-m+2}),\dots,F^{1}\circ\textnormal{ad}^{1}_{x_{i}}(x_{n})\bigr{)},$

or the (row) matrix of the form $F^{1}\circ\textnormal{ad}^{1}_{x_{i}}$ on
$\mathcal{G}^{1}$ with respect to the basis $\\{x_{n-m+1},\dots,x_{n}\\}$.
Therefore, we can connect the rank of $M_{F}$, as well as the dimension of
$\Omega_{F}$, with the maximum number of linearly independent matrices in the
set

$\\{\textnormal{ad}^{1}_{x_{1}},\textnormal{ad}^{1}_{x_{2}},\dots,\textnormal{ad}^{1}_{x_{n-m}}\\}.$

Remark that the vector space spanned by the above set and the vector space
spanned by the set
$\mathcal{S}:=\\{\textnormal{ad}^{1}_{x}:x\in\mathcal{L}\oplus_{\rho}\mathcal{G}^{1}\\}$
coincide. It follows that

(3.16) $\dim\Omega_{F}=2\operatorname*{rank}{M_{F}}=2\dim
F^{1}(\langle\mathcal{S}\rangle),$

where $\langle\mathcal{S}\rangle$ denotes the vector space spanned by $S$, and
$F^{1}(\langle\mathcal{S}\rangle)$ denotes the vector space $\\{F^{1}\circ
f:f\in\langle\mathcal{S}\rangle\\}$. In the following theorem, we will use the
maximum number mentioned above, which is equal to the dimension of
$\langle\mathcal{S}\rangle$, to determine some interest properties of
$\mathcal{G}$, including the decomposition.

######  Theorem 3.7.

Let $\mathcal{G}$ satisfy the hypothesis of Theorem 3.3 and let $\mathcal{S}$
be the set of adjoint actions of all points in $\mathcal{G}$, i.e.
$\mathcal{S}=\\{\textnormal{ad}^{1}_{x}:x\in\mathcal{G}\\}$. Denote by $m$ the
dimension of $\mathcal{G}^{1}$ and by $r$ the dimension of the vector space
spanned by $\mathcal{S}$. Then the following assertions hold.

  1. 1.

$k\leq 2r\leq 2(n-m)$,

  2. 2.

$r=n-m$ if and only if $\mathcal{G}$ is indecomposable,

  3. 3.

$2r=k$ if and only if every non-zero element in the vector space spanned by
$\mathcal{S}$ is an automorphism on the vector space $\mathcal{G}^{1}$.

######  Proof 3.8.

Followed by Theorem 3.3, $\mathcal{G}$ is a direct sum of two commutative
subalgebras $\mathcal{L}$ and $\mathcal{G}^{1}$. By the second assertion in
that theorem, we can fix a basis $\\{x_{1},\ldots,x_{n}\\}$ of $\mathcal{G}$
so that

(3.17) $\left\\{\begin{array}[]{ll}\mathcal{L}=&\langle
x_{1},\ldots,x_{n-m}\rangle\\\ \mathcal{G}^{1}=&\langle
x_{n-m+1},\ldots,x_{n}\rangle\end{array}\right.,$

and $\textnormal{ad}^{1}_{x_{1}}$ is an automorphism on the vector space
$\mathcal{G}^{1}$.

  1. 1.

Since $\mathcal{G}$ is spanned by $\\{x_{1},x_{2},\dots,x_{n}\\}$, the real
vector space spanned by $\mathcal{S}$, $\langle\mathcal{S}\rangle$, is also
spanned by $\\{\textnormal{ad}^{1}_{x_{i}}:i=1,2,\ldots,n\\}$. Moreover, the
commutation of $\mathcal{G}^{1}$ implies that
$\textnormal{ad}^{1}_{x_{i}}=\boldsymbol{0}$ for every $n-m+1\leq i\leq n$.
Hence,

(3.18) $r\leq n-m.$

On the other hand, in the light of the equation (3.16), we easily see that,
for any $F\in\mathcal{G}^{*}$ which does not vanish on $\mathcal{G}^{1}$,

(3.19) $0\neq\dim\Omega_{F}=2\dim F^{1}(\langle\mathcal{S}\rangle)\leq
2\dim\langle\mathcal{S}\rangle=2r.$

The inequalities (3.18) and (3.19) establish the formula in the first item.

  2. 2.

By Theorem 3.1, we conclude that $\mathcal{G}$ is decomposable if and only if
$\mathcal{G}=\mathcal{K}\oplus\mathbb{R}$ for some sub-algebra
$\mathcal{K}\subseteq\mathcal{G}$. Hence, $\mathcal{G}$ is decomposable if and
only if there is an element $x\in\mathcal{G}\setminus\mathcal{G}^{1}$ so that
it is contained in the center of $\mathcal{G}$. Because
$x\notin\mathcal{G}^{1}$, we get

(3.20) $x=y+z$

for some $0\neq y\in\mathcal{L}$ and for some $z\in\mathcal{G}^{1}$. Since $x$
belongs to the center of $\mathcal{G}$ and $\mathcal{G}^{1}$ is commutative,
we obtain

(3.21) $\textnormal{ad}^{1}_{y}=\boldsymbol{0}.$

Therefore, $\mathcal{G}$ is decomposable if and only if the dimension of the
vector space spanned by

$\\{\textnormal{ad}^{1}_{x_{1}},\ldots,\textnormal{ad}^{1}_{x_{n-m}}\\}$

is strictly less than $n-m$, or $r<n-m$, which is our assertion.

  3. 3.

In light of the equation (3.16), what we need to show is the following
statement: $\dim
F^{1}(\langle\mathcal{S}\rangle)=\dim(\langle\mathcal{S}\rangle)$ for every
$F\in(\mathcal{G})^{*}$ with $F^{1}\neq 0$ if and only if every non-zero
element of $\langle\mathcal{S}\rangle$ is an automorphism on the vector space
$\mathcal{G}^{1}$. Equivalently, $\dim
F^{1}(\langle\mathcal{S}\rangle)<\dim(\langle\mathcal{S}\rangle)$ for some
$F\in(\mathcal{G})^{*}$ with $F^{1}\neq 0$ if and only if there is a non-zero
element $f$ of $\langle\mathcal{S}\rangle$ which is not an automorphism on the
vector space $\mathcal{G}^{1}$.

Indeed, the existence of a non-zero element $f\in\langle\mathcal{S}\rangle$
which is not an automorphism on the vector space $\mathcal{G}^{1}$ is
equivalent to the existence of an element $0\neq x\in\mathcal{G}^{1}$ so that
$f(x)=0$, hence is equivalent to the existence of an element
$F\in(\mathcal{G})^{*}$ so that

(3.22) $\left\\{\begin{array}[]{ll}F^{1}\neq\boldsymbol{0},\\\ F^{1}\circ
f=\boldsymbol{0}.\end{array}\right.$

If the equation (3.22) holds, then by choosing a basis for
$\langle\mathcal{S}\rangle$ which contains $f$, we can see easily that the
dimension of $F^{1}(\langle\mathcal{S}\rangle)$ is strictly less than the
dimension of $\langle\mathcal{S}\rangle$.

Conversely, if $\dim
F^{1}(\langle\mathcal{S}\rangle)<\dim(\langle\mathcal{S}\rangle)$, then there
is a basis $\\{f_{1},f_{2},\dots,f_{r}\\}$ of $\langle\mathcal{S}\rangle$ so
that

(3.23) $F^{1}(\alpha f_{1}+\cdots+\alpha_{r}f_{r})=\boldsymbol{0}$

for some non-zero $(\alpha_{1},\dots,\alpha_{r})\in\mathbb{R}^{r}$. This
proves the existence of a non-zero element $f\in\langle\mathcal{S}\rangle$ so
that the equation (3.22) holds. The proof is completed.

To close this subsection, let’s present the connection between 2-step MD-
algebras with 1-step MD-algebras in the next theorem.

######  Theorem 3.9.

Let $\mathcal{G}$ be an MD${}_{k}(n)$-algebra. Then $\dim\mathcal{G}^{2}\leq
n-k$. Furthermore, the quotient Lie algebra
$\mathcal{H}:=\mathcal{G}/\mathcal{G}^{2}$ is also an MDk-algebra.

######  Proof 3.10.

The assertion of the theorem is obviously true when $\mathcal{G}^{2}$ is
trivial. Therefore, we only need to consider the case
$\dim\mathcal{G}^{2}=p>0$. Denote by $m$ the dimension of $\mathcal{G}^{1}$
($m>p$). Let’s fix a basis $\\{x_{1},x_{2},\dots,x_{n}\\}$ of $\mathcal{G}$ so
that $\mathcal{G}^{1}=\langle x_{n-m+1},\dots,x_{n}\rangle$ and
$\mathcal{G}^{2}=\langle x_{n-p+1},\dots,x_{n}\rangle$. Since
$\mathcal{G}^{2}$ is an ideal of $\mathcal{G}$,
$x^{*}_{n-m+1}([x_{i},x_{j}])=0$ for every $i,j$ with $1\leq i\leq n$ and
$n-p+1\leq j\leq n$. Therefore, by setting $F=x^{*}_{n-m+1}$ we get

(3.24) $B_{F}=\left[\begin{array}[]{cl}M&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}_{p\times p}\end{array}\right]$

for some square matrix $M$ of order $n-p$. It follows that the rank of $B_{F}$
is at most $n-p$. On the other hand, because $\mathcal{G}$ is an
$\text{MD}_{k}(n)$-algebra and $F^{1}\neq 0$, we have

(3.25) $\dim\Omega_{F}=k.$

It follows from Proposition 2.3 that

(3.26) $k=\dim\Omega_{F}=\operatorname*{rank}B_{F}\leq n-p.$

In other words,

(3.27) $\dim\mathcal{G}^{2}\leq n-k.$

This proves the first part of the theorem.
To prove the second part, we still assume $\\{x_{1},x_{2},\dots,x_{n}\\}$ is a
basis of $\mathcal{G}$ so that $\\{x_{n-m+1},\dots,x_{n}\\}$ and
$\\{x_{n-p+1},\dots,x_{n}\\}(p<m)$ are bases of $\mathcal{G}^{1}$ and
$\mathcal{G}^{2}$, respectively. It is standard to check that

(3.28)
$x_{q}^{*}([x_{i},x_{j}])=\overline{x_{q}^{*}}([x_{i}+\mathcal{G}^{2},x_{j}+\mathcal{G}^{2}])\text{
for any }1\leq i,j\leq n,\text{ and }1\leq q\leq n-p,$

where $\overline{x_{q}^{*}}$ is the corresponding element of $x_{q}^{*}$ in
$(\mathcal{H}^{1})^{*}$. Therefore, the dimensions of the coadjoint orbits
$\Omega_{\overline{F}}$ (of $\mathcal{H}$) and $\Omega_{F}$ (of $\mathcal{G}$)
are equal for any $F\in(\mathcal{G})^{*}$ with
$\overline{F}|_{\mathcal{H}^{1}}\neq 0$. This completes the proof.

###  3.2 The complete classification of all
MD$\boldsymbol{{}_{n-1}(n)}$-algebras

Now, we will state the last main result of the paper, which gives the complete
classification of all indecomposable MD${}_{n-1}(n)$-algebras for $n\geq 4$.

######  Theorem 3.11.

Let $\mathcal{G}$ be an indecomposable MD${}_{n-1}(n)$-algebra with $n\geq 4$.
Then $\mathcal{G}$ is isomorphic to one of the followings:

  1. 1.

The real Heisenberg Lie algebra

$\mathfrak{h}_{2m+1}=\langle x_{i},y_{i},z:i=1,\dots,m\rangle,$

where non-zero Lie brackets are given by $[x_{i},y_{i}]=z$ for every $1\leq
i\leq m$.

  2. 2.

The Lie algebra

$\mathfrak{s}_{5,45}=\langle x_{1},x_{2},y_{1},y_{2},z\rangle,$

where non-zero Lie brackets are given by

$[x_{1},y_{1}]=y_{1},[x_{1},y_{2}]=y_{2},[x_{1},z]=2z,[x_{2},y_{1}]=y_{2},[x_{2},y_{2}]=-y_{1},[y_{1},y_{2}]=z.$

######  Proof 3.12.

The proof will be divided into 3 cases.

  * •

Case 1. Assume $\dim\mathcal{G}^{1}=1$. Followed by the classifications of
solvable real Lie algebras having 1-dimensional derived ideal in [21], we
easily see that $\mathcal{G}$ is isomorphic to $\mathfrak{h}_{2m+1}$ with
$2\leq m\in\mathbb{N}$ because $\mathcal{G}$ is indecomposable and $n\geq 4$.

  * •

Case 2. Assume that $\mathcal{G}^{1}$ is commutative and
$\dim\mathcal{G}^{1}\geq 2$. We will show that this case is excluded.

We first claim that $\dim\mathcal{G}$ is exactly five. Indeed, according to
Theorem 3.3, one can see that
$\mathcal{G}=\mathcal{L}\oplus_{\rho}\mathcal{G}^{1}$ where $\mathcal{L}$ is a
commutative subalgebra of $\mathcal{G}$ with
$\mathcal{L}\cap\mathcal{G}^{1}=\\{0\\}$. Hence, we can choose a basis
$\mathfrak{b}=\left\\{x_{1},x_{2},\dots,x_{n}\right\\}$ of $\mathcal{G}$ so
that

(3.29) $\left\\{\begin{array}[]{rl}\mathcal{G}^{1}=&\langle
x_{n-m+1},x_{n-m+2},\dots,x_{n}\rangle\\\ \mathcal{L}=&\langle
x_{1},x_{2},\dots,x_{n-m}\rangle\end{array}\right..$

Because both $\mathcal{G}^{1}$ and $\mathcal{L}$ are commutative, we have

(3.30) $[x_{i},x_{j}]=0\quad\text{if }\left[\begin{array}[]{c}1\leq i,j\leq
n-m\\\ n-m+1\leq i,j\leq n\end{array}\right..$

On the other hand, the commutation of $\mathcal{G}^{1}$ implies that
$\\{\textnormal{ad}^{1}_{x}:x\in\mathcal{L}\\}$ is a family of commuting
endomorphisms [Proposition 2.6]. As a consequence of the second item of
Proposition 2.7, we can assume that the matrix of
$\textnormal{ad}^{1}_{x_{i}}$ with respect to the basis $\mathfrak{b}$ is of
block upper triangular form in which each block is of size up to 2, for every
$1\leq i\leq n-k$. With this assumption in hand, the equality (3.30) implies
that

(3.31) $x_{n}^{*}([x_{i},x_{j}])=0\quad\text{if }\left[\begin{array}[]{c}1\leq
i,j\leq n-2\\\ n-1\leq i,j\leq n\\\ \end{array}\right..$

Therefore,

(3.32) $0\neq\dim\Omega_{x_{n}^{*}}=\operatorname*{rank}B_{x_{n}^{*}}\leq 4$

Hence, $n-1=\dim\Omega_{x_{n}^{*}}\leq 4$, or $n\leq 5.$ Since
$\dim\Omega_{F}$ is even for every $F\in\mathcal{G}^{*}$ and $n\geq 4$, $n$
must be equal to $5$.

To the best of our knowledge, all 1-step solvable real Lie algebras of
dimension 5 are completely classified by Jacqueline Dozias [9, 19]. In
particular, Vu L. A. and K. P. Shum gave the classification of 1-step solvable
MD-algebras of dimension 5 in [22]. According to their classification, there
is no indecomposable 1-step solvable MD-algebra of dimension 5 so that its
maximal dimension of $K$-orbits is 4. Therefore, this case is excluded.

  * •

Case 3. Assume that $\mathcal{G}^{1}$ is non-commutative. If so,
$\dim\mathcal{G}^{1}\geq 2$ and $\dim\mathcal{G}^{2}\geq 1$. It follows from
Theorem 3.9 that $\dim\mathcal{G}^{2}\leq\dim\mathcal{G}-\dim\Omega_{F}=1$ for
every $0\neq F\in(\mathcal{G}^{1})^{*}$. Hence, $\dim\mathcal{G}^{2}=1$ and
$\mathcal{H}=\mathcal{G}/\mathcal{G}^{2}$ is a solvable real Lie algebra in
which $\dim\mathcal{H}=n-1=\dim\Omega_{F}$ for every $0\neq
F\in(\mathcal{G}^{1})^{*}$. In other word, $\mathcal{H}$ is an indecomposable
SMD-algebra. It follows from [20, Theorem 1] that $n-1=\dim\mathcal{H}\leq 4$.
Hence, $n\leq 5$. Once again, by using the classification of 5-dimensional Lie
algebras in [9, 19], especially of those in which the first derived ideal is
non-commutative in [21], we get exactly one 5-dimensional Lie algebra
satisfying the requirement which is defined in item 2. Remark that this
algebra is denoted as $\mathfrak{s}_{5,45}$ in [19] and the proof is complete.

######  Remark 3.13.

To sharpen Theorem 3.11, using Theorem 3.1, we complete the paper by the
following remarks.

  1. (1)

All solvable Lie algebras of dimension 3 are obviously MD-algebras. Therefore,
except $\mathbb{R}^{3}$ (the real 3-dimensional abelian Lie algebra), all the
remaining solvable non-abelian ones are MD${}_{2}(3)$-algebras. One can find
their classification in [19].

  2. (2)

Any decomposable MD${}_{n-1}(n)$-algebra is a direct sum of $\mathbb{R}$ and
an indecomposable non-abelian SMD-algebra of dimension $n-1$. Therefore, by
applying directly the classification of SMD-algebras in [20, Theorem 1], we
easily see that any decomposable MD${}_{n-1}(n)$-algebra is isomorphic to
either $\textnormal{aff}(\mathbb{R})\oplus\mathbb{R}$ for $n=3$ or
$\textnormal{aff}(\mathbb{C})\oplus\mathbb{R}$ for $n=5$ where

     * •

$\textnormal{aff}(\mathbb{R})=\langle x,y\rangle$ is defined by $[x,y]=y$;

     * •

$\textnormal{aff}(\mathbb{C})=\langle x_{1},x_{2},y_{1},y_{2}\rangle$ is
defined by $[x_{1},x_{2}]=[y_{1},y_{2}]=0$,
$[x_{1},y_{1}]=y_{1},[x_{1},y_{2}]=y_{2}$, and
$[x_{2},y_{1}]=y_{2},[x_{2},y_{2}]=-y_{1}.$

Acknowledgment. The first author would like to express his deep gratitude to
his wife - Thuy Nguyen Thanh for her encouragement and support.

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