1 Introduction

Leibniz algebras are a non-antisymmetric generalization of Lie algebras. They were introduced in 1965 by Bloh [7], who called them D-algebras, and in 1993 Loday [13] made them popular and studied their (co)homology.

Definition 1

An algebra \((L,[-,-])\) over a field \(\mathbb {F}\) is called a Leibniz algebra if for any \(x,y,z\in L\), the so-called Leibniz identity

$$\begin{aligned} \big [x,[y,z]\big ]=\big [[x,y],z\big ]-\big [[x,z],y\big ] \end{aligned}$$

holds.

Since first works about Leibniz algebras around 1993 several researchers have tried to find analogs of important theorems in Lie algebras. For instance, the classical results on Cartan subalgebras [1, 15], Engel’s theorem [3], Levi’s decomposition [5], properties of solvable algebras with given nilradical [10] and others from the theory of Lie algebras are also true for Leibniz algebras.

Namely, an analogue of Levi’s decomposition for Leibniz algebras asserts that any Leibniz algebra is decomposed into a semidirect sum of its solvable radical and a semisimple Lie algebra. Therefore, the main problem of the description of finite-dimensional Leibniz algebras consists of the study of solvable Leibniz algebras. In turn, the classification of all solvable Leibniz algebras has been reduced to the classification of the nilpotent Leibniz algebras since middle of the twentieth century. Partially due to the difficulties of obtaining a complete classification of the nilpotent Leibniz algebras, the study of nilpotent Leibniz algebras must be accompanied by imposing additional conditions such as constraints on the index of nilpotency of the algebra, on the characteristic sequence and grading.

Given an arbitrary Leibniz algebra L we define the lower central series:

$$\begin{aligned} L^1 = L, \quad L^{k+1} = [L^k,L], \quad k \ge 1. \end{aligned}$$

A Leibniz algebra L is called nilpotent if there exists \(s \in \mathbb {N}\) such that

$$\begin{aligned} L^1 \supset L^2 \supset \dots \supset L^s=\{0\}. \end{aligned}$$

The smallest number s is called the index of nilpotency or nilindex of L. The index of nilpotency of an n-dimensional algebra is at most \(n + 1\). Among the nilpotent Leibniz algebras, we distinguish the null-filiform (nilindex \(n+1\)) and filiform (nilindex n) algebras. The concept of null-filiform makes no sense for Lie algebras.

An n-dimensional Leibniz algebra L is called null-filiform if \(\dim L^i = (n+1)-i, \ 1 \le i \le n + 1\).

The null-filiform Leibniz algebras are those that have the maximal index of nilpotency, i.e., \(n+1\), and, in each dimension, up to isomorphism, there exists a unique complex null-filiform Leibniz algebra (see [4]).

An n-dimensional Leibniz algebra L is called filiform if \(\dim L^i=n-i\), \(2 \le i \le n\).

Let L be a finite-dimensional nilpotent Leibniz algebra. Set

$$\begin{aligned} L_i := L^i/L^{i+1}, \quad 1 \le i \le s-1, \end{aligned}$$

where s is the nilindex of the algebra L, and denote

$$\begin{aligned} \hbox {gr} L:= L_1\oplus L_2\oplus \dots \oplus L_{s-1}. \end{aligned}$$

Since \([L_i,L_j] \subseteq L_{i+j}\), we obtain a graded algebra \(\hbox {gr} L\). The grading constructed above will be called the natural grading. If a Leibniz algebra G is isomorphic to the algebra \(\hbox {gr} L\), then G is called a naturally graded Leibniz algebra. In [4], complex n-dimensional naturally graded filiform non-Lie Leibniz algebras are classified, and it is proved that there are only two non-isomorphic Leibniz algebras.

In fact, each non-Lie Leibniz algebra L contains a non-trivial ideal (later denoted by I and usually called Leibniz kernel), which is generated by the squares of the elements of the algebra L, i.e., \(I= \langle \{[x,x] \mid x \in L\}\rangle \). Moreover, it is easy to see that this ideal belongs to the right annihilator of L, that is \([L,I]=0\). Note also that the ideal I is the minimal ideal with the property that the quotient algebra L/I is a Lie algebra. The ideal I plays an important role in the theory since it determines the (possible) non-Lie character of L. Observe that we can write \(L = (L / I) \oplus I\) as direct sum of vector spaces. The quotient algebra L/I is said to be the corresponding Lie algebra to the Leibniz algebra L. In fact a Leibniz algebra L is called simple when \([L,L] \ne I\) and its only ideals are {0}, I and L. If L is a simple Leibniz algebra, then L/I is a simple Lie algebra, but the reciprocal is not true. However, a Leibniz algebra L is semisimple if and only if the Lie algebra L/I is semisimple.

One of the approaches in the investigation on Leibniz algebras is the description of these algebras such that their corresponding Lie algebras are a given Lie algebra (see [2, 9, 16, 17]).

The map \(I \times (L / I) \rightarrow I\) defined as \((v,\overline{x}) \mapsto [v,x]\), \(v \in I, \, x \in L\), endows I with a structure of right (L / I)-module. If we consider the direct sum of vector spaces \((L / I) \oplus I\), then the structure of right (L / I)-module defines a Leibniz algebra structure on \((L / I) \oplus I\), known as the hemisemidirect product of L/I with I (see [12]), with the following multiplication

$$\begin{aligned}&\left[ \overline{x}+v,\overline{y}+w\right] := \left[ \overline{x},\overline{y}\right] + [v,y], \quad \text {that is},\\&\left[ \overline{x},\overline{y}\right] = \overline{[x,y]}, \quad [v,\overline{x}] = [v,x], \quad \left[ \overline{x},v\right] = 0, \quad [v,w] = 0, \quad x, y \in L, \ v,w \in I. \end{aligned}$$

In fact, this structure of Leibniz algebra is isomorphic to the initial one of L.

Therefore, for a given Lie algebra G and a right G-module M, we can construct a Leibniz algebra \(L=G\oplus M\) by the above construction.

Let \(\mathfrak {D}\) be the four-dimensional Diamond Lie algebra with basis \(\{\overline{1},\overline{x},\frac{\overline{\partial }}{\partial x},\overline{e}\}\). In the paper [17] it is constructed the so-called Fock module over \(\mathfrak {D}\), the linear space \({\mathbb {F}}[x]\) of polynomials on x, where \(\mathbb {F}\) is an algebraically closed field of characteristic zero. The linear space \({\mathbb {F}}[x]\) is called the Fock\(\mathfrak {D}\)-module if an action \({\mathbb {F}}[x] \times \mathfrak {D} \rightarrow {\mathbb {F}}[x]\) is defined as follows:

$$\begin{aligned} \begin{array}{lll} (p(x),\overline{1})&{} \mapsto &{} p(x),\\ (p(x),\overline{x})&{} \mapsto &{} xp(x),\\ (p(x),\frac{\overline{\partial }}{\partial x})&{} \mapsto &{}\frac{\partial }{\partial x}(p(x)),\\ (p(x),\overline{e}) &{} \mapsto &{} -x\frac{\partial (p(x))}{\partial x}, \end{array} \end{aligned}$$

for any \(p(x) \in \mathbb {F}[x]\).

Furthermore, in the paper [17] infinite-dimensional Leibniz algebras with corresponding Lie algebra is the complex Diamond Lie algebra \(\mathfrak {D}\) and the ideal I is the Fock \(\mathfrak {D}\)-module are classified.

Theorem 1

[17]] A Leibniz algebra L with conditions \(L/I\cong \mathfrak {D}\), and I is the Fock \(\mathfrak {D}\)-module, admits a basis

$$\begin{aligned} \left\{ \overline{1}, \overline{x}, \frac{\overline{\partial }}{\partial x}, \ \overline{e}, \ x^{t} \mid t\in \mathbb {N}\cup \{0\}\right\} , \end{aligned}$$

such that the multiplication table in this basis has the following form:

$$\begin{aligned} \left\{ \begin{aligned}&\,[\overline{e},\quad \overline{x}] =\overline{x}, \quad [\overline{x}, \overline{e}] =-\overline{x}, \left[ \overline{e},\frac{\overline{\partial }}{\partial x}\right] =-\frac{\overline{\partial }}{\partial x}, \quad \left[ \frac{\overline{\partial }}{\partial x},\overline{e}\right] =\frac{\overline{\partial }}{\partial x}, \quad \left[ \overline{x},\frac{\overline{\partial }}{\partial x}\right] =\overline{1}, \\&\left[ \frac{\overline{\partial }}{\partial x},\overline{x}\right] =-\overline{1}, \, [x^t,\overline{1}] = x^{t}, \quad [x^t,\overline{x}] = x^{t+1}, \quad \left[ x^{t},\frac{\overline{\partial }}{\partial x}\right] = tx^{t-1}, \,\, [x^t,\overline{e}]= -tx^{t}, \end{aligned}\right. \end{aligned}$$

where the omitted products are equal to zero.

Now, we give the definition of the general Diamond Lie algebras.

The real general Diamond Lie algebra\(\mathfrak {D}_m\) is a \((2m+2)\)-dimensional Lie algebra with basis

$$\begin{aligned} \{J,P_1,P_2,\dots ,P_m,Q_1,Q_2,\dots ,Q_m,T\} \end{aligned}$$

and nonzero relations

$$\begin{aligned}{}[J,P_k]=Q_k, \qquad [J,Q_k]=-P_k, \qquad [P_k,Q_k]=T, \qquad 1\le k\le m. \end{aligned}$$

The complexification of the Diamond Lie algebra is \(\mathfrak {D}_m \otimes _\mathbb {R} \mathbb {C}\), for which we shall keep the same symbol \(\mathfrak {D}_m(\mathbb {C})\), and it has the following (complex) basis:

$$\begin{aligned} P_k^{+} = P_k - iQ_k, \qquad Q_k^{-} = P_k + iQ_k, \qquad T, \qquad J, \qquad 1\le k\le m, \end{aligned}$$

where i is the imaginary unit, and with nonzero commutators

$$\begin{aligned}{}[J,P_k^{+}] = iP_k^{+}, \qquad [J,Q_k^{-}] = - iQ_k^{-}, \qquad [P_k^{+},Q_k^{-}] = 2iT, \qquad 1\le k\le m. \end{aligned}$$
(1)

The Ado’s theorem in Lie Theory states that every finite-dimensional complex Lie algebra can be represented as a matrix Lie algebra, formed by matrices. However, that result does not specify which is the minimal dimension of the matrices involved in such representations. In [8], the value of the minimal dimension of the matrices for abelian Lie algebras and Heisenberg algebras \(\mathfrak {h}_m\), defined on a \((2m + 1)\)-dimensional vector space with basis \(\{X_1, \dots , X_m, Y_1, \dots , Y_m, Z\}\), and brackets \([X_i, Y_i] = Z\), is found. For abelian Lie algebras of dimension n the minimal dimension is \(\lceil 2 \sqrt{n-1} \rceil \).

Lemma 1

[8] For the Heisenberg Lie algebras \(\mathfrak {h}_m\), the minimal faithful matrix representation has dimension equal to \(m+2\).

The theory of representations of Leibniz algebras was introduced in [14]. Let L be a Leibniz algebra and V a vector space over the field \(\mathbb {F}\). A representation of the Leibniz algebra L on the vector space V is a pair \((\lambda , \rho )\) of linear maps \(\lambda , \rho :L \rightarrow \mathfrak {gl}(V)\) satisfying the following properties:

$$\begin{aligned} \rho _{[x,y]}&=\rho _y \rho _x - \rho _x \rho _y, \\ \lambda _{[x,y]}&=\rho _y \lambda _x - \lambda _x \rho _y,\\ \lambda _{[x,y]}&=\rho _y \lambda _x + \lambda _x \lambda _y, \qquad \text {for all} \, x,y\in L, \, \text {where} \, \lambda _x=\lambda (x) \, \text {and} \, \rho _x=\rho (x). \end{aligned}$$

If \(\mathfrak {g}\) is a Lie algebra, then its Lie representation \(\varphi :\mathfrak {g} \rightarrow \mathfrak {gl}(V)\) becomes a Leibniz representation with \(\lambda =\varphi \) and \(\rho = - \varphi \), or also by taking \(\lambda =0\) and \(\rho =-\varphi \).

Notice that the concepts of representations of Lie algebras and Leibniz algebras are different. The Ado’s theorem on the existence of faithful representations is a relevant theorem in the theory of Lie algebras. The analogous in the case of Leibniz algebras was proved in [6] in an easier way and gives a stronger result. That is because the kernel of the Leibniz algebra representation is the intersection of the kernels of \(\lambda \) and \(\rho \), \(\ker (\lambda , \rho ) := \ker \lambda \cap \ker \rho = \{ x \in L \mid \lambda _x= 0=\rho _x\}\), which are in general different, in contrast to the representations of Lie algebras, where these kernels are the same. Therefore, a faithful representation of Leibniz algebras can be obtained more easily than a faithful representation in the case of Lie algebras. The representation theories of Leibniz algebras and that their corresponding Lie algebras are very related. In fact, in [11] the authors use representations of the Lie algebra L/I to construct representations of the Leibniz algebra L.

In this paper we find a minimal faithful representation of the \((2m+2)\)-dimensional complex general Diamond Lie algebra, \(\mathfrak {D}_m(\mathbb {C})\), which is isomorphic to a subalgebra of the special linear Lie algebra \(\mathfrak {sl}(m+2,\mathbb {C})\). Moreover, we find a faithful representation of \(\mathfrak {D}_m\) which is isomorphic to a subalgebra of the symplectic Lie algebra \(\mathfrak {sp}(2m+2,\mathbb {R})\). Then we construct Leibniz algebras with corresponding general Diamond Lie algebra and the ideal generated by the squares of elements in these faithful representations.

2 Leibniz Algebras Constructed by a Minimal Faithful Representation of the General Diamond Lie Algebras

In this section we are going to study Leibniz algebras L such that \(L/ I \cong \mathfrak {D}_m(\mathbb {C})\) and the \(\mathfrak {D}_m(\mathbb {C})\)-module I is a minimal faithful representation, that is, the action \(I \times \mathfrak {D}_m(\mathbb {C}) \rightarrow I\) gives rise to a minimal faithful representation of \(\mathfrak {D}_m(\mathbb {C})\). Moreover, this representation factorizes through \(\mathfrak {sl}(m+2,\mathbb {C})\).

Proposition 1

Let \(\mathfrak {D}_m(\mathbb {C})\) be a \((2m+2)\)-dimensional general Diamond Lie algebra with basis

$$\begin{aligned} \{J,P_1^+,P_2^+,\dots ,P_m^+,Q_1^-,Q_2^-,\dots ,Q_m^-,T\}. \end{aligned}$$

Then its minimal faithful representation is given by the correspondence

Proof

Consider the linear map \(\varphi :\mathfrak {D}_m(\mathbb {C})\rightarrow \mathfrak {sl}({m+2},\mathbb {C})\) given by

$$\begin{aligned} \varphi (J)&= \frac{im}{m+2}e_{1,1}-\sum \limits _{s=2}^{m+1}\frac{2i}{m+2}e_{s,s}+\frac{im}{m+2}e_{m+2,m+2},\qquad \varphi (T) = -\frac{i}{2}e_{1,m+2},\\ \varphi (P_k^+)&=e_{1,m+2-k}, \qquad \varphi (Q_k^-)=e_{m+2-k,m+2}, \qquad 1 \le k \le m, \end{aligned}$$

where \(e_{i,j}\) is the matrix with (i, j)-th entry 1 and all others 0.

By checking \([\varphi (x),\varphi (y)]=\varphi (x)\varphi (y)-\varphi (y)\varphi (x)\) for all \(x,y \in \mathfrak {D}_m(\mathbb {C})\), we verify that \(\varphi \) is an injective homomorphism of Lie algebras. It is easy to see that \(\mathfrak {D}_m(\mathbb {C})/ \mathbb {C}J \cong \mathfrak {h}_m\). By Lemma 1 we obtain that it is minimal. \(\square \)

Let us denote by \(V=\mathbb {C}^{m+2}\) the natural \(\varphi (\mathfrak {D}_m(\mathbb {C}))\)-module and endow it with a \(\mathfrak {D}_m(\mathbb {C})\)-module structure, \(V \times \mathfrak {D}_m(\mathbb {C}) \rightarrow V\), given by

$$\begin{aligned} (x, e) := x \varphi (e), \end{aligned}$$

where \(x \in V\) and \(e\in \mathfrak {D}_m(\mathbb {C})\).

Then the action of \(\mathfrak {D}_m(\mathbb {C})\) on \(V=\langle X_1,X_2,\dots ,X_{m+2} \rangle \) is given as follows:

$$\begin{aligned} \left\{ \begin{aligned}&(X_1,J) = \frac{im}{m+2} X_1, \\&(X_k,J) = -\frac{2i}{m+2} X_k,&2 \le k \le m+1, \\&(X_{m+2},J) = \frac{im}{m+2} X_{m+2}, \\&(X_1, P_k^+) = X_{m+2-k},&1 \le k \le m,\\&(X_{m+2-k},Q_{k}^-) = X_{m+2},&1 \le k \le m,\\&(X_1,T) = -\frac{i}{2}X_{m+2}, \end{aligned}\right. \end{aligned}$$
(2)

and the remaining products in the action being zero.

Now we investigate Leibniz algebras L such that \(L/I \cong \mathfrak {D}_m(\mathbb {C})\) and \(I = V\) as a \(\mathfrak {D}_m(\mathbb {C})\)-module.

Theorem 2

Let L be an arbitrary Leibniz algebra with corresponding Lie algebra \(\mathfrak {D}_m(\mathbb {C})\) and the ideal I associated as \(\mathfrak {D}_m(\mathbb {C})\)-module defined by (2). Then there exists a basis

$$\begin{aligned} \{J,P_1^+,P_2^+,\dots ,P_m^+,Q_1^-,Q_2^-,\dots ,Q_m^-,T,X_1,X_2,\dots ,X_{m+2}\} \end{aligned}$$

of L such that

$$\begin{aligned}{}[\mathfrak {D}_m(\mathbb {C}),\mathfrak {D}_m(\mathbb {C})]\subseteq \mathfrak {D}_m(\mathbb {C}). \end{aligned}$$

Proof

Here we shall use the multiplication table (1) of the complex Diamond Lie algebra. Let us assume that

$$\begin{aligned}{}[J,J]=\sum \limits _{k=1}^{m+2}\delta _kX_k. \end{aligned}$$

Then by setting

$$\begin{aligned} {J^\prime } := J+\frac{i(m+2)\delta _1}{m}X_1-\sum \limits _{k=2}^{m+1}\frac{i(m+2)\delta _i}{2}X_i+\frac{i(m+2)\delta _{m+2}}{m}X_{m+2}, \end{aligned}$$

we can assume that \([J,J]=0\).

Let us denote

$$\begin{aligned}{}[J,P_k^+]=iP_k^++\sum \limits _{s=1}^{m+2}\alpha _{k,s}X_s, \quad [J,Q_k^-]=-iQ_k^-+\sum \limits _{s=1}^{m+2}\beta _{k,s}X_s, \quad 1\le k\le m. \end{aligned}$$

Taking the following basis transformation:

$$\begin{aligned}&J^{\prime }=J, \quad P_k^{+\prime }=P_k^+-\sum \limits _{s=1}^{m+2}i\alpha _{k,s}X_s, \quad Q_k^{-\prime }=Q_k^-+\sum \limits _{k=2}^{m+2}i\beta _{k,s}X_s, \\&\quad T'=-i/2[P_1^{+\prime },Q_1^{-\prime }], \quad 1\le k\le m, \end{aligned}$$

we can assume that

$$\begin{aligned}{}[J,P_k^+]=iP_k^+, \quad [J,Q_k^-]=-iQ_k^-, \quad [P_1^+,Q_1^-]=2iT, \quad 1\le k\le m. \end{aligned}$$

By applying the Leibniz identity to the triples \(\{J,J,P_k^+\}, \ \{J,J,Q_k^-\}\), we derive

$$\begin{aligned}{}[P_k^+,J]=-[J,P_k^+], \qquad [Q_k^-,J]=-[J,Q_k^-], \qquad 1\le k\le m. \end{aligned}$$

By considering Leibniz identity for the triples we have the following constraints.

\(\square \)

3 Leibniz Algebras Constructed by a Faithful Representation of the General Diamond Algebra Which is Isomorphic to a Subalgebra of \(\mathfrak {sp}(2m+2,\mathbb {R})\)

In this section we are going to study Leibniz algebras L such that \(L/ I \cong \mathfrak {D}_m\) and the \(\mathfrak {D}_m\)-module I is a faithful representation. Moreover, this representation factorizes through \(\mathfrak {sp}(2m+2,\mathbb {R})\).

Proposition 2

Let \(\mathfrak {D}_m\) be a \((2m+2)\)-dimensional real general Diamond Lie algebra with the basis \(\{J,P_1,P_2,\dots ,P_m,Q_1,Q_2,\dots ,Q_m,T\}\). Then it is isomorphic to a subalgebra of \(\mathfrak {sp}(2m+2,\mathbb {R})\) via the map

Proof

Consider the linear map \(\varphi :\mathfrak {D}_m \rightarrow \mathfrak {sp}(2m+2,\mathbb {R})\) given by

$$\begin{aligned} \varphi (J)&= -\sum \limits _{s=2}^{m+1}e_{k,2m+3-k}+\sum \limits _{s=m+2}^{2m+1}e_{k,2m+3-k},\qquad \varphi (T) = 2e_{1,2m+2},\\ \varphi (P_k)&=e_{1,1+k}-e_{2m+2-k,2m+2}, \quad \varphi (Q_k)=e_{1,2m+2-k}+e_{k+1,2m+2}, \quad 1 \le k \le m. \end{aligned}$$

It is easy to check that \(\varphi \) is an injective homomorphism of Lie algebras.\(\square \)

By using the same techniques that in the previous section, we obtain that the action of \(\mathfrak {D}_m\) on \(V=\langle X_1,X_2,\dots ,X_{m+2} \rangle \) is given by

$$\begin{aligned} \left\{ \begin{aligned} (X_k,J)&= - X_{2m+3-k},&2 \le k \le m+1,\\ (X_k,J)&= X_{2m+3-k},&m+2 \le k \le 2m+1,\\ (X_1,P_k)&= X_{k+1},&1 \le k \le m,\\ (X_{2m+2-k},P_k)&= - X_{2m+2},&1 \le k \le m,\\ (X_1,Q_k)&= X_{2m+2-k},&1 \le k \le m,\\ (X_{k+1},Q_k)&= X_{2m+2},&1 \le k \le m, \\ (X_1,T)&= 2X_{2m+2},&1 \le k \le m, \end{aligned}\right. \end{aligned}$$
(3)

and the remaining products in the action being zero.

Theorem 3

An arbitrary real Leibniz algebra with corresponding Lie algebra \(\mathfrak {D}_m\), and the ideal I associated as \(\mathfrak {D}_m\)-module defined by (3), admits a basis

$$\begin{aligned} \{J,P_1,P_2,\dots ,P_m,Q_1,Q_2,\dots ,Q_m,T,X_1,X_2,\dots ,X_{2m+2}\} \end{aligned}$$

such that the multiplication table \([\mathfrak {D}_m,\mathfrak {D}_m]\) has the following form:

$$\begin{aligned} \left\{ \begin{array}{ll} {[}J,J]=a_1X_{2m+2}, &{} [J,P_k]=-[P_k,J]=Q_k,\\ {[}J,Q_k]=-[Q_k,J]=-P_k, &{} [P_k,Q_k]=-[Q_k,P_k]=T,\\ {[}P_k,P_s]=[Q_k,Q_s]=b_{k,s}X_{2m+2}, &{} [P_k,Q_s]=[Q_k,P_s]=c_{k,s}X_{2m+2},\\ \end{array}\right. \end{aligned}$$

with the restrictions

$$\begin{aligned} b_{k,s}=-b_{s,k}, \qquad c_{k,s}=c_{s,k}, \end{aligned}$$

where \(1\le k,s\le m, \ k\ne s\).

Proof

Let us assume that

$$\begin{aligned}{}[J,J]=\sum \limits _{k=1}^{m+2}\delta _kX_k \quad \text { and } \quad [J,T]=\sum \limits _{k=1}^{2m+2}\rho _kX_k. \end{aligned}$$

By using change of bases and the Leibniz identity, we get

$$\begin{aligned}{}[J,J]=\delta _1X_1+\delta _{2m+2}X_{2m+2} \quad \text { and } \quad [J,T]=\rho _1X_1. \end{aligned}$$

Let us suppose

$$\begin{aligned}{}[J,P_k]=Q_k+\sum \limits _{s=1}^{2m+2}\lambda _{k,s}X_s, \quad [J,Q_k]=-P_k+\sum \limits _{s=1}^{2m+2}\mu _{k,s}X_s, \quad 1\le k\le m. \end{aligned}$$

Taking the following basis transformation:

$$\begin{aligned}&J^{\prime }=J, \quad P_k^{\prime }=P_k-\sum \limits _{s=1}^{2m+2}\mu _{k,s}X_s, \quad Q_k^{\prime }=Q_k+\sum \limits _{k=1}^{2m+2}\lambda _{k,s}X_s, \quad \\&T^{\prime }=[P_1^{\prime },Q_1^{\prime }], \quad 1\le k\le m, \end{aligned}$$

we can assume that

$$\begin{aligned}{}[J,P_k]=Q_k, \qquad [J,Q_k]=-P_k, \qquad [P_1,Q_1]=T, \qquad 1\le k\le m. \end{aligned}$$

By applying the Leibniz identity to the triples \(\{J,J,P_k\}, \ \{J,J,Q_k\}, \ \{J,P_k,T\}, \)\( \{J,Q_k,T\}\) and \(\{P_1,T,Q_1\}\), we derive

$$\begin{aligned} {[}P_k,J]= & {} -[J,P_k], \qquad [Q_k,J]=-[J,Q_k], \qquad 1\le k\le m,\\ {[}Q_k,T]= & {} \rho _1X_{k+1}, \qquad [P_k,T]=-\rho _1X_{2m+2-k}, \quad 1\le k\le m,\quad [T,T]=0 \end{aligned}$$

We set

$$\begin{aligned} \left\{ \begin{array}{ll} {[}P_j,Q_j]=T+\sum \limits _{t=1}^{2m+2}\beta _{j,t}X_t, &{}[Q_k,P_k]=-T+\sum \limits _{t=1}^{2m+2}\gamma _{k,t}X_t, \\ {[}P_k,P_s]=\sum \limits _{t=1}^{2m+2}\eta _{k,s,t}X_t, &{}[Q_k,Q_s]=\sum \limits _{t=1}^{2m+2}\theta _{k,s,t}X_t, \\ {[}P_k,Q_s]=\sum \limits _{t=1}^{2m+2}\nu _{k,s,t}X_t, \ k\ne s, &{}[Q_k,P_s]=\sum \limits _{t=1}^{2m+2}\xi _{k,s,t}X_t, \ k\ne s, \end{array}\right. \end{aligned}$$

where \( 2\le j\le m, \ 1\le k,s\le m\).

By applying the Leibniz identity we get

$$\begin{aligned}&[P_k,P_s]=-[Q_s,Q_k], \quad [P_k,Q_s]=[P_s,Q_k], \quad [Q_k,P_s]=[Q_s,P_k], \nonumber \\&\quad 1\le k,s\le m, \ k\ne s. \end{aligned}$$
(4)

By applying the Leibniz identity to \(\{P_1,J,Q_1\}\) and \(\{P_1,P_1,Q_1\}\), we have

$$\begin{aligned}{}[T,J]=\sum \limits _{s=2}^{2m+2}2\eta _{1,1,s}X_s, \qquad [T,P_1]=\frac{3}{2}\rho _1X_{2m+1}+\eta _{1,1,2}X_{2m+2}. \end{aligned}$$

Next, by using the Leibniz identity we derive

$$\begin{aligned} \left\{ \begin{aligned}{}[T,J]&=0,&[Q_1,P_1]&=-T, \\ [P_1,P_1]&=\frac{1}{2}\rho _1X_1,&[Q_1,Q_1]&=\frac{1}{2}\rho _1X_1, \\ [T,P_1]&=\frac{3}{2}\rho _1X_{2m+1},&[T,Q_1]&=-\frac{3}{2}\rho _1X_{2},\\ [P_k,Q_k]&=-[Q_k,P_k]=T,&[P_k,P_k]&=[Q_k,Q_k]=\frac{1}{2}\rho _1X_1, \\ [T,P_k]&=\frac{3}{2}\rho _1X_{2m+2-k},&[T,Q_k]&=-\frac{3}{2}\rho _1X_{k+1}, \end{aligned}\right. \end{aligned}$$

where \(2\le k\le m\).

By verifying the Leibniz identity on elements, we obtain the following restrictions.

By applying the Leibniz identity to \(\{P_k,P_s,J\}\) and \(\{Q_k,Q_s,J\}\), we get

$$\begin{aligned}&\xi _{k,s,2m+2}=-\nu _{k,s,2m+2}, \quad \theta _{k,s,t}=-\eta _{k,s,t},&1\le k,s\le m, \ 2\le t\le 2m+1, \ k\ne s, \\&\nu _{k,s,t}+\xi _{k,s,t}=-\eta _{k,s,2m+3-t}&\qquad 2\le t\le m+1,\\&\nu _{k,s,t}+\xi _{k,s,t}=\eta _{k,s,2m+3-t}&\qquad m+2\le t\le 2m+1. \end{aligned}$$

Let us consider the identity

$$\begin{aligned}{}[[Q_k,P_s],J]=[Q_k,[P_s,J]]+[[Q_k,J],P_s]=-[Q_k,Q_s]+[P_k,P_s]. \end{aligned}$$

We have that \(\theta _{k,s,2m+2}=\eta _{k,s,2m+2}\) and

$$\begin{aligned} \begin{array}{ll} \xi _{k,s,t}=-2\eta _{k,s,2m+3-t},&{} \qquad 2\le t\le m+1,\\ \xi _{k,s,t}=2\eta _{k,s,2m+3-t},&{} \qquad m+2\le t\le m+1, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ll} \nu _{k,s,t}=\eta _{k,s,2m+3-t}, &{} \qquad 2\le t\le m+1,\\ \nu _{k,s,t}=-\eta _{k,s,2m+3-t}, &{} \qquad m+2\le t\le 2m+1. \end{array} \end{aligned}$$

Analogously, by applying the Leibniz identity to the triple \(\{P_k,Q_s,J\}\), we get

$$\begin{aligned} \begin{array}{ll} \nu _{k,s,t}=-2\eta _{k,s,2m+3-t},&{} \qquad 2\le t\le m+1,\\ \nu _{k,s,t}=2\eta _{k,s,2m+3-t},&{} \qquad m+2\le t\le 2m+1. \end{array} \end{aligned}$$

We get that \(\nu _{k,s,t}=0, \ 1\le k,s\le m, \ 2\le t\le 2m+1, \ k\ne s\). It implies that \(\eta _{k,s,t}=\xi _{k,s,t}=\theta _{k,s,t}=0\) for \(1\le k,s\le m, \ 2\le t\le 2m+1, \ k\ne s\).

Hence, we have

$$\begin{aligned} \begin{array}{ll} {[}P_k,P_s]=[Q_k,Q_s]=\eta _{k,s,2m+2}X_{2m+2}, &{} \qquad 1\le k,s\le m, \quad k\ne s, \\ {[}P_k,Q_s]=[Q_k,P_s]=\nu _{k,s,2m+2}X_{2m+2}, &{} \qquad 1\le k,s\le m, \quad k\ne s. \end{array} \end{aligned}$$

By equation (4) we have the following restrictions

$$\begin{aligned} \eta _{k,s,2m+2}=-\eta _{s,k,2m+2}, \qquad \nu _{k,s,2m+2}=\nu _{s,k,2m+2}, \qquad 1\le k,s\le m, \ k\ne s. \end{aligned}$$

Finally, we apply the Leibniz identity to the triple \(\{P_k,P_k,P_s\}\) with \(k\ne s\) and we obtain \(\rho _1=0\). We denote again \((\delta _{2m+2},\eta _{k,s,2m+2},\nu _{k,s,2m+2})=(a_1,b_{k,s},c_{k,s})\).\(\square \)