\section{Introduction}
In \cite{kap80:01} A.~Kaplan introduced $H$-type Lie algebras, 
which belong to a special class of 2-step nilpotent Lie algebras. 
This class is associated with the Clifford algebra for an inner product space 
and an admissible module of the Clifford algebra. 
An $H$-type Lie algebra obtained by replacing the inner product to a 
general scalar product first appeared in \cite{cia00:0}. 
This Lie algebra with the scalar product is called a pseudo $H$-type Lie 
algebra, which is exactly defined below. 

Let $\mathfrak n$ be a finite dimensional 2-step nilpotent real Lie algebras, 
that is, 
$\mathfrak n$ is a finite dimensional real Lie algebra satisfying 
$[\mathfrak n,\mathfrak n]\ne0$ and 
$[\mathfrak n,[\mathfrak n,\mathfrak n]]=0$. 
Let $\innerproduct$ be a scalar product on $\mathfrak n$ 
such that the center $\mathfrak n_{-2}$ of $\mathfrak n$ is a 
non-degenerate subspace of $(\mathfrak n,\innerproduct)$. 
Here a scalar product on $\mathfrak n$ means a non-degenerate symmetric 
bilinear form on $\mathfrak n$. 
Let $\mathfrak n_{-1}$ be the orthogonal complement of $\mathfrak n_{-2}$ with respect to $\innerproduct$. 
The pair $(\mathfrak n,\innerproduct)$ is called a pseudo $H$-type Lie algebra  if for any $z\in \mathfrak n_{-2}$ the endomorphism $J_z$ of $\mathfrak n_{-1}$ defined by 
$\inph{J_z(x)}{y}=\inph{z}{[x,y]}$ $(x,y\in\mathfrak n_{-1})$ 
satisfies the Clifford condition $J_z^2=-\inph{z}{z}1_{\mathfrak n_{-1}}$, 
where $1_{\mathfrak n_{-1}}$ is the identity transformation of 
$\mathfrak n_{-1}$. 
In particular, if $\innerproduct$ is positive definite, then 
$(\mathfrak n,\innerproduct)$ is simply called an $H$-type Lie algebra. 

Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra. 
Then $\mathfrak n=\mathfrak n_{-2}\oplus \mathfrak n_{-1}$ becomes a 
non-degenerate fundamental graded Lie algebra of the second kind, 
which is called associated with  $(\mathfrak n,\innerproduct)$. 

Now we explain the notion of a fundamental graded Lie algebra and its prolongation briefly. 
A finite dimensional graded Lie algebra (GLA) 
$\mathfrak m=\bigoplus\limits_{p<0}\mathfrak g_p$ is called a fundamental graded Lie algebra (FGLA) of 
the $\mu$-th kind if the following conditions hold: 
(i) $\mathfrak g_{-1}\ne0$, 
and $\mathfrak m$ is generated by $\mathfrak g_{-1}$; 
(ii) $\mathfrak g_p=0$ for all $p<-\mu$, 
where $\mu$ is a positive integer. 
Furthermore an FGLA $\mathfrak m=\bigoplus\limits_{p<0}\mathfrak g_p$ is 
called non-degenerate if for $x\in\mathfrak g_{-1}$, $[x,\mathfrak g_{-1}]=0$ 
implies $x=0$. 
For a given FGLA $\mathfrak m=\bigoplus\limits_{p<0}\mathfrak g_p$ 
there exists a GLA $\cgla g$ satisfying the following conditions: 
(P1) The negative part $\check{\mathfrak g}_-=\bigoplus\limits_{p<0}\check{\mathfrak g}_p$ of 
$\cgla g$ coincides with a given FGLA $\mathfrak m$ as a GLA; 
(P2) For $x\in \check{\mathfrak g}_{p}$ $(p\geqq0)$, $[x,\mathfrak g_{-1}]=0$ 
implies $x=0$; 
(P3) $\cgla g$ is maximum among GLAs satisfying the conditions (P1) and (P2) 
above. 
The GLA $\cgla g$ is called the (Tanaka) prolongation of the FGLA 
$\mathfrak m$. 
Given the prolongation $\cgla g$ of an FGLA $\mathfrak m$, 
an element $E$ of $\check{\mathfrak g}_0$ is called the characteristic element of $\cgla g$ if $[E,x]=px$ for all $x\in\check{\mathfrak g}_p$ and $p\in\mathbb Z$. Also $\ad(\check{\mathfrak g}_0)|\mathfrak m$ is a subalgebra  of $\Der(\mathfrak m)$ isomorphic to $\check{\mathfrak g}_0$; 
we identify it with $\check{\mathfrak g}_0$ in what follows, 
so that $D\in\check{\mathfrak g}_0$ is identified with $\ad(D)|\mathfrak m$. 
(For the details of FGLAs and a construction of the 
prolongation, see \cite[\S5]{tan70:1}). 

For a given pseudo $H$-type Lie algebra $(\mathfrak n,\innerproduct)$ 
the prolongation $\cgla g$ of the FGLA $\mathfrak n$ is finite dimensional if and only if $\dim \mathfrak n_{-2}\geqq3$ (\cite[Theorem 2.4, and Propositions 4.4 and 4.5]{AS14:0}). 
Moreover in \cite[Theorem 3.1]{AS14:1} A.~Altomani and A.~Santi proved that 
if $\dim \mathfrak n_{-2}\geqq3$ and the prolongation is not trivial 
(i.e., $\check{\mathfrak g}_1\ne0$),  
then $\cgla g$ is a finite dimensional SGLA 
(In this paper we abbreviate simple GLA to SGLA). 

We next give the notion of a conformal pseudo-subriemannian FGLA and 
its prolongation. 
We say that the pair $(\mathfrak m,[g])$ of a real FGLA $\mathfrak m$ of 
the $\mu$-th kind $(\mu\geqq2)$ and the conformal class $[g]$ of 
a scalar product $g$ on $\mathfrak g_{-1}$ is a conformal pseudo-subriemannian 
FGLA (cps-FGLA). 
Let $\cgla g$ be the prolongation of $\mathfrak m$, and let 
 $\mathfrak g_0$ be the subalgebra of $\check{\mathfrak g}_0$ consisting 
of all the elements $D$ of $\check{\mathfrak g}_0$ 
such that $\ad(D)|\mathfrak g_{-1}\in\mathfrak{co}(\mathfrak g_{-1},g)$.  
We define a sequence $(\mathfrak g_p)_{p\geqq1}$ inductively as follows: 
$l$ being a positive integer, suppose that we defined $\mathfrak g_1,\dots,\mathfrak g_{l-1}$ as subspaces of 
$\check{\mathfrak g}_1,\dots,\check{\mathfrak g}_{l-1}$ 
respectively, in such a way that $[\mathfrak g_p,\mathfrak g_r]\subset\mathfrak g_{p+r}$ $(0<p<l,r<0)$. Then we define $\mathfrak g_l$ to be the subspace of 
$\check{\mathfrak g}_l$ consisting of all the elements $D$ of $\check{\mathfrak g}_l$ such that $[D,\mathfrak g_r]\subset\mathfrak g_{l+r}$ $(r<0)$. 
If we put $\gla g$, then it becomes a graded subalgebra of $\cgla g$, 
which is called the prolongation of $(\mathfrak m,\mathfrak g_0)$. 
The prolongation of $(\mathfrak m,\mathfrak g_0)$ is also called that of the cps-FGLA $(\mathfrak m,[g])$. 
The prolongation $\gla g$ of the cps-FGLA $(\mathfrak m,[g])$ is finite dimensional. 
If $\gla g$ is semisimple, then the cps-FGLA $(\mathfrak m,[g])$ is said to 
be of semisimple type. 
In the previous paper \cite{yat18:01} we classified the prolongations of cps-FGLAs of semisimple type. 

Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra. 
The pair $(\mathfrak n,[\innerproduct_{-1}])$ becomes a cps-FGLA, 
which is called associated with 
$(\mathfrak n,\innerproduct)$. 
Here we denote by $\innerproduct_{k}$ 
the restriction of $\innerproduct$ to $\mathfrak n_k$. 

In \cite{KS17:0} A.~Kaplan and M.~Sublis introduced the notion of a 
$\Div H$-type Lie algebra (or a Lie algebra of type $\Div H$) 
and classified the finite dimensional real SGLAs whose negative parts are 
isomorphic to some $\Div H$-type Lie algebra. 
In \cite{KS16:0} they also proved that the prolongation of the FGLA associated with an $H$-type Lie algebra is not trivial if and only 
if it is a $\Div H$-type Lie algebra. 
In \S3, inspired by the studies in \cite{KS17:0} and \cite{gom96:0}, we give a little generalization of a $\Div H$-type Lie algebra, 
which is called a pseudo $\Div H$-type Lie algebra. 
More precisely, the pseudo $\Div H$-type Lie algebras consist of 
three classes (pseudo $\Div H$-type Lie algebras of the first, 
the second and the third classes). 
We determine the prolongations of the FGLAs associated with 
pseudo $\Div H$-type Lie algebras by an elementary method. 
It is known that a pseudo $H$-type Lie algebra satisfying the $J^2$-condition 
becomes a pseudo $\Div H$-type Lie algebra of the first class, and vice versa (cf.\cite{MKM18:0}). 
In \S4 we prove that a pseudo $H$-type Lie algebra 
satisfies the $J^2$-condition if and only if the prolongation of the associated cps-FGLA is a finite dimensional SGLA (Theorem \ref{th41}). 

By \cite[Theorem 3.1]{AS14:1} and \cite[Theorem 5.3]{KS14:0}, 
the prolongation $\gla g$ of the cps-FGLA associated with a pseudo $H$-type Lie algebra 
$(\mathfrak n,\innerproduct)$ is a finite dimensional SGLA of real rank one 
if the following conditions hold: (i) $\mathfrak g_1\ne0$; (ii) $\innerproduct_{-1}$ is definite.  
However if $\innerproduct_{-1}$ is indefinite, 
$\mathfrak g$ has a more complicated form. 
In \S5 we show that if 
$\mathfrak g_2\ne0$, then $\gla g$ is a finite dimensional SGLA and coincides 
with the prolongation of $\mathfrak n$ under the additional condition 
``$\dim \mathfrak n_{-2}\geqq3$'' (Theorem \ref{th53}). 

In \cite{FMM17:0} K.~Furutani et al. investigated the prolongations of the FGLAs associated with pseudo $H$-type Lie algebras. 
From their results, we conjecture that if the prolongation of the FGLA associated with a pseudo $H$-type Lie algebra is not trivial, 
then it is of pseudo $\Div H$-type. 


\section{Pseudo $H$-type Lie algebras}
Following \cite{cia00:0} we define pseudo $H$-type Lie algebras. 
Let $\mathfrak n$ be a finite dimensional 2-step nilpotent real Lie algebra 
equipped with a non-degenerate symmetric bilinear form 
$\inph{\cdot}{\cdot}$ on $\mathfrak n$. 
The pair $(\mathfrak n,\inph{\cdot}{\cdot})$ is called a pseudo 
$H$-type Lie algebra if the following conditions hold: 
\begin{enumerate}
\renewcommand{\labelenumi}{(H.\arabic{enumi}) }
\item 
The restriction  of $\innerproduct$ to 
the center $\mathfrak n_{-2}$ of $\mathfrak n$ is non-degenerate. 
\item Let $\mathfrak n_{-1}$ be the orthogonal complement of the center $\mathfrak n_{-2}$ of $\mathfrak n$ with respect to $\innerproduct$. 
For any $z\in \mathfrak n_{-2}$ the endomorphism  $J_{z}$ of 
$\mathfrak n_{-1}$ defined by 
\begin{equation}\inph{J_z(x)}{y} =\inph{z}{[x,y]} \qquad 
x,y\in \mathfrak n_{-1},  
\label{eq22}
\end{equation}
satisfies the following condition 
\begin{equation}
J_z^2=-\inph{z}{z} 1_{\mathfrak n_{-1}},
\label{eq23}
\end{equation}  
where $1_{\mathfrak n_{-1}}$ is the identity transformation of $\mathfrak n_{-1}$. 
\end{enumerate}
The condition \eqref{eq23} is called the Clifford condition. 
In particular if $\innerproduct$ is positive definite, 
then $(\mathfrak n,\innerproduct)$ is simply called an $H$-type Lie algebra. 
Given a pseudo $H$-type Lie algebra $(\mathfrak n,\innerproduct)$ 
we can easily see that: 
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})\ }
\item For any $z\in \mathfrak n_{-2}$ 
the linear mapping $J_z$ is skew-symmetric; 
\item $\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2}$ is 
a non-degenerate FGLA of the second kind. 
\end{enumerate}
The FGLA $\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2}$ 
is called associated with 
the pseudo $H$-type Lie algebra $(\mathfrak n,\langle\cdot\mid\cdot\rangle)$.  
The pair $(\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},
[\innerproduct_{-1}])$ becomes a conformal pseudo-subriemannian FGLA 
(cps-FGLA), which is called associated with 
the pseudo $H$-type Lie algebra $(\mathfrak n,\langle\cdot\mid\cdot\rangle)$. 
Given two pseudo $H$-type Lie algebras $(\mathfrak n,\innerproduct)$ and 
$(\mathfrak n',\innerproduct')$, we say that 
$(\mathfrak n,\innerproduct)$ is isomorphic to  
$(\mathfrak n',\innerproduct')$ if there exists a Lie algebra isomorphism $\varphi$ of 
$\mathfrak n$ onto $\mathfrak n'$ such that $\varphi$ is an isometry of  
$(\mathfrak n,\innerproduct)$ onto $(\mathfrak n',\innerproduct')$. 
Moreover we say that 
$(\mathfrak n,\innerproduct)$ is equivalent to  
$(\mathfrak n',\innerproduct')$ if 
there exists a Lie algebra isomorphism $\varphi$ of 
$\mathfrak n$ onto $\mathfrak n'$ such that: 
(i) $\varphi(\mathfrak n_{-1})=\mathfrak n'_{-1}$, and 
$\varphi|\mathfrak n_{-1}$ is an isometry or an anti-isometry of 
$(\mathfrak n_{-1},\innerproduct_{-1})$ onto $(\mathfrak n'_{-1},\innerproduct'_{-1})$; 
(ii) $\varphi|\mathfrak n_{-2}$ is an isometry of 
$(\mathfrak n_{-2},\innerproduct_{-2})$ onto 
$(\mathfrak n'_{-2},\innerproduct'_{-2})$. 
If a pseudo $H$-type Lie algebra $(\mathfrak n,\innerproduct)$ 
is equivalent to a pseudo $H$-type Lie algebra 
$(\mathfrak n',\innerproduct')$, 
then the prolongation of $(\mathfrak n,[\innerproduct_{-1}])$ is 
isomorphic to that of  $(\mathfrak n',[\innerproduct'_{-1}])$. 
\begin{lemma}
Let $(\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},\innerproduct)$ 
be a pseudo $H$-type Lie algebra. 
We define a new scalar product $\innerproduct'$ on $\mathfrak n$ as follows: 
$$\inph{x}{y}'=\alpha\inph{x}{y}\ (x,y\in\mathfrak n_{-1}),\quad
\inph{z}{w}'=\beta\inph{z}{w}\ (z,w\in\mathfrak n_{-2}),\quad
\inph{\mathfrak n_{-1}}{\mathfrak n_{-2}}'=0,$$
where $\alpha,\beta$ are nonzero real numbers. 
The pair $(\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},\innerproduct')$ also becomes a pseudo $H$-type Lie algebra if and only if $\alpha^2=\beta$. 
In this case, 
the cps-FGLA associated with $(\mathfrak n,\innerproduct')$ is 
$(\mathfrak n,[\alpha \innerproduct_{-1}])$. 
\label{lem24}
\end{lemma}
\begin{proof}
By \eqref{eq22}, 
for $x,y\in\mathfrak n_{-1}$ and $z\in\mathfrak n_{-2}$, 
$\inph{\alpha^{-1}\beta J_{z}(x)}{y}'=\beta\inph{J_{z}(x)}{y}
=\beta\inph{z}{[x,y]}=\inph{z}{[x,y]}'$. 
By \eqref{eq23}, 
$(\alpha^{-1}\beta J_{z})^2=\alpha^{-2}\beta^2 J_{z}^2=
-\alpha^{-2}\beta^2\inph{z}{z}1_{\mathfrak n_{-1}}=-\alpha^{-2}\beta\inph{z}{z}'1_{\mathfrak n_{-1}}$. This proves the first statement. 
The last statement is clear. 
\end{proof}

The proof of the following lemma is due to the proof of 
\cite[Theorem 2]{FM17:0}. 
\begin{lemma} 
Let $(\mathfrak n^{(1)},\innerproduct^{(1)})$ 
and $(\mathfrak n^{(2)},\innerproduct^{(2)})$ be pseudo $H$-type Lie algebras. 
Assume that there exists a GLA isomorphism $\varphi$ of $\mathfrak n^{(1)}$ onto $\mathfrak n^{(2)}$. 
Then there exists a GLA isomorphism $\psi$ of $\mathfrak n^{(1)}$ onto $\mathfrak n^{(2)}$ and a positive real number $\alpha$ 
such that: {\rm (i)} $\psi|\mathfrak n_{-2}^{(1)}$ is an isometry or an anti-isometry; 
{\rm (ii)} $\psi|\mathfrak n_{-1}^{(1)}=\alpha\varphi|\mathfrak n_{-1}^{(1)}$.  
\label{lem26}
\end{lemma}
\begin{remark} Let $(\mathfrak n^{(1)},\innerproduct^{(1)})$  and $(\mathfrak n^{(2)},\innerproduct^{(2)})$ be $H$-type Lie algebras. 
If $\mathfrak n^{(1)}$ is isomorphic to $\mathfrak n^{(2)}$ as a GLA, then
$(\mathfrak n^{(1)},\innerproduct^{(1)})$ is isomorphic to 
$(\mathfrak n^{(2)},\innerproduct^{(2)})$ as an $H$-type Lie algebra
(\cite[Theorem 2]{KS16:0}). 
\end{remark}

\begin{proposition}
Let $\gla g$ be a finite dimensional real SGLA 
such that the negative part $\mathfrak g_-=\bigoplus_{p<0}\limits\mathfrak g_p$ is an FGLA of the second kind. 
Let $\innerproduct^{(i)}$ $(i=1,2)$ be scalar products on $\mathfrak g_-$. 
Assume that: 
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi}) }
\item $(\mathfrak g_-,\innerproduct^{(1)})$ and 
$(\mathfrak g_-,\innerproduct^{(2)})$ are pseudo $H$-type Lie algebras 
whose associated FGLAs coincide with $\mathfrak g_-$ as a GLA. 
\item For $i=1,2$ the prolongation of the associated csp-GLA 
$(\mathfrak g_-,[\innerproduct_{-1}^{(i)}])$ coincides with $\mathfrak g$. 
\end{enumerate}
Then 
\begin{enumerate}
\item $[\innerproduct_{-1}^{(1)}]$ is equal to $[\innerproduct_{-1}^{(2)}]$ or 
$[-\innerproduct_{-1}^{(2)}]$; 
\item $[\innerproduct_{-2}^{(1)}]=[\innerproduct_{-2}^{(2)}]$, 
 
\end{enumerate} 
Consequently, $(\mathfrak g_-,\innerproduct^{(1)})$ is equivalent to 
$(\mathfrak g_-,\innerproduct^{(2)})$.  
\label{prop22}
\end{proposition}
\begin{proof} 
Let $\varphi$ be the identity transformation of $\mathfrak g_-$. 
By the assumption (i) $\varphi$ is a GLA isomorphism of $\mathfrak g_-$ onto itself. 
By Lemma \ref{lem26}, there exists a GLA isomorphism $\psi$ of 
$\mathfrak g_-$ onto itself such that: (i) the restriction $\psi|\mathfrak g_{-2}$ to 
$\mathfrak g_{-2}$ of $\psi$ is an isometry or an anti-isometry; 
(ii) there exist a nonzero real number $\alpha'$ 
such that 
$\psi|\mathfrak g_{-2}=\alpha'^2\varphi|\mathfrak g_{-2}$ and 
$\psi|\mathfrak g_{-1}=\alpha'\varphi|\mathfrak g_{-1}$.  
Hence $\alpha'^4\inph{\cdot}{\cdot}^{(2)}_{-2}
=\pm\inph{\cdot}{\cdot}^{(1)}_{-2}$. 
By assumptions (ii), (iii) and \cite[Proposition 5.2]{yat18:01}, 
$\inph{\cdot}{\cdot}^{(2)}_{-1}$ coincides with 
$\inph{\cdot}{\cdot}^{(1)}_{-1}$ multiplied by a nonzero real number. 
By Lemma \ref{lem24}, 
there exists a nonzero real number $\alpha$ such that 
$\inph{\cdot}{\cdot}^{(2)}_{-1}=\alpha
\inph{\cdot}{\cdot}^{(1)}_{-1}$, 
$\inph{\cdot}{\cdot}^{(2)}_{-2}=\alpha^2
\inph{\cdot}{\cdot}^{(1)}_{-2}$. 
Thus assertions (i) and (ii) are proved. 
We define a linear mapping $f$ of $\mathfrak g_-$ into itself as follows: 
$$f(x)=|\alpha|^{-1/2}x\quad (x\in \mathfrak g_{-1}), \qquad
f(z)=|\alpha|^{-1} z \quad (z\in \mathfrak g_{-2});$$ 
then $f$ is a GLA isomorphism and we see that 
$$\begin{aligned}
& \inph{f(x)}{f(y)}^{(2)}=|\alpha|^{-1}\inph{x}{y}^{(2)}=
\sgn(\alpha)\inph{x}{y}^{(1)}\quad (x,y\in\mathfrak g_{-1}), \\
& \inph{f(z)}{f(z')}^{(2)}=|\alpha|^{-2}\inph{z}{z'}^{(2)}=
\inph{z}{z'}^{(1)}\quad (z,z'\in\mathfrak g_{-2}). 
\end{aligned}$$
Hence 
$(\mathfrak g_-,\innerproduct^{(1)})$ is equivalent to 
$(\mathfrak g_-,\innerproduct^{(2)})$. 
\end{proof}
\section{Pseudo $\Div H$-type Lie algebras}
In this section we introduce pseudo $\Div H$-type Lie algebras. 
The pseudo $\Div H$-type Lie algebras consist of 
pseudo $\Div H$-type Lie algebras 
$\mathfrak H^{(1)}(\mathbb F,S)$ of the first class, 
$\mathfrak H^{(2)}(\mathbb F,S,\gamma)$ of the second class, 
and $\mathfrak H^{(3)}(\mathbb F,S)$ of the third class, 
which is defined below. 

\subsection{Cayley algebras}
Let $\mathbb F$ be $\mathbb C$, $\mathbb C'$, $\mathbb H$, $\mathbb H'$, $\mathbb O$ or $\mathbb O'$, where 
$\mathbb C$ (resp. $\mathbb C'$, $\mathbb H$, $\mathbb H'$, $\mathbb O$, $\mathbb O'$) is a Cayley algebra of the complex numbers 
(resp. the split complex numbers, the Hamilton's quaternions, 
the split quaternions, the Cayley's octonions, the split octonions). 
Here we consider $\mathbb F$ as an algebra over $\mathbb R$. 
We denote by $\mathbb F(\gamma)$ the Cayley extension of $\mathbb F$ defined by $\gamma$, where $\gamma=\pm1$ (cf. \cite[Ch.3, no.5]{bou98:1}). Namely $\mathbb F(\gamma)$ is an algebra over 
$\mathbb R$ which $\mathbb F(\gamma)=\mathbb F\times\mathbb F$ as a module and the multiplication on $\mathbb F(\gamma)$ is defined by 
$$(x_1,x_2)(y_1,y_2)
=(x_1y_1+\gamma \overline{y_2}x_2,x_2\overline{y_1}+y_2x_1).$$ 
Clearly $\mathbb F\times\{0\}$ is a subalgebra of $\mathbb F(\gamma)$ isomorphic to $\mathbb F$; we shall identify it with $\mathbb F$ in what follows, 
so that $x\in\mathbb F$ is identified with $(x,0)$. 
Let $\ell=(0,1)$, so that $(x,y)=x+y\ell$ for $x,y\in\mathbb F$. 
Note that: 
(i) $\ell\alpha=\overline{\alpha}\ell$; 
(ii) $\alpha(\beta \ell)=(\beta\alpha)\ell$; 
(iii) $(\alpha \ell)\beta=(\alpha\bar{\beta})\ell$; 
(iv) $(\alpha \ell)(\beta \ell)=\gamma(\overline{\beta}\alpha)$; 
(v) $\ell^2=\gamma$, 
where $\alpha,\beta\in \mathbb F$. 
When $\mathbb F=\mathbb H$ (resp. $\mathbb F=\mathbb H'$) 
we put $\mathbb F_0=\mathbb C$, and 
$\gamma_0=-1$ 
(resp. $\gamma_0=1)$; then 
$\mathbb F=\mathbb F_0(\gamma_0)$. Let $\ell_0$ be the element of $\mathbb F$ corresponding to 
the element $(0,1)\in\mathbb F_0(\gamma_0)=\mathbb F_0\times \mathbb F_0$. 
We denote by $\mathbb F^c=\mathbb F\oplus\sqrt{-1}\mathbb F$, 
$\mathbb F(\gamma)^c=\mathbb F(\gamma)\oplus\sqrt{-1}\mathbb F(\gamma)$ the complexifications 
of $\mathbb F$, $\mathbb F(\gamma)$ respectively. 
Let $\pr_1$ and $\pr_2$ be the projections of 
$\mathbb F(\gamma)^c=\mathbb F^c\times \mathbb F^c$ onto $\mathbb F^c$ 
defined by $\pr_i(x_1,x_2)=x_i$ $(i=1,2)$. 
Note that $\pr_1(\overline{\alpha})=\overline{\pr_1(\alpha)}$, 
$\pr_2(\overline{\alpha})=-\pr_2(\alpha)$, 
$\pr_1(\ell\alpha)=\gamma\overline{\pr_2(\alpha)}$, 
$\pr_2(\ell\alpha)=\overline{\pr_1(\alpha)}$, 
where $\alpha\in \mathbb F(\gamma)^c$. 
We define a mapping $R$ of $\mathbb F(\gamma)^c$ to $\mathbb R$ 
by $R(u+\sqrt{-1}v)=\rea(u)$ $(u,v\in\mathbb F(\gamma))$. 
For $z\in \mathbb F=\mathbb F\times\{0\}$ and $\alpha\in \mathbb F(\gamma)^c$ we obtain $R(z\pr_1(\alpha))=R(z\alpha)$. 
We extend the conjugation ``$\overset{-}{\cdot}$'' on $\mathbb F(\gamma)$ 
to $\mathbb F(\gamma)^c$ by 
$\overline{u+\sqrt{-1}v}=\overline{u}+\sqrt{-1}\overline{v}$. 
\subsection{Pseudo $\Div H$-type Lie algebras of the first class}
\label{sec32}

Let $\mathbb F$ be $\mathbb C$, $\mathbb C'$, $\mathbb H$, $\mathbb H'$, 
$\mathbb O$ or $\mathbb O'$. Let $S$ be a  real symmetric matrix of order $n$ 
such that $S^2=1_n$, where  $1_n$ is the identity matrix of order $n$. 
We put 
$$\mathfrak n_{-1}=\mathbb F^n,\quad \mathfrak n_{-2}=\ima \mathbb F, 
\quad \mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},$$
where we assume $n=1$ in case $F=\mathbb O$ or $\mathbb O'$. 
Note that $\mathbb F^n$ is the set of all the $\mathbb F$-valued row vectors of order $n$. 
We define a bracket operation on $\mathfrak n$ as follows: 
$$[x,y]=-2\ima(xSy^*)=ySx^*-xSy^* \quad (x,y\in \mathfrak n_{-1}), 
\quad [\mathfrak n_{-1},\mathfrak n_{-2}]=[\mathfrak n_{-2},\mathfrak n_{-2}]=0;$$ 
then $(\mathfrak n,[\cdot,\cdot])$ becomes an FGLA of the second kind. 
Furthermore we define a symmetric bilinear form $\langle \cdot\mid \cdot\rangle$ on $\mathfrak n$ 
as follows: 
$$\begin{aligned}
&
\langle x\mid y\rangle=2\rea(xSy^*)\quad (x,y\in\mathfrak n_{-1}), \\
& \langle z\mid w\rangle=\rea(z\overline{w})=-\rea(zw)\quad (z,w\in\mathfrak n_{-2}),
\quad \langle \mathfrak n_{-1}\mid \mathfrak n_{-2}\rangle=0. 
\end{aligned}$$
The linear mapping $J_z$ defined by \eqref{eq23} has the following form: 
$J_z(x)=-zx$. 
Thus $(\mathfrak n,\langle\cdot\mid\cdot\rangle)$ becomes a pseudo $H$-type Lie algebra, which is denoted by 
$\mathfrak H^{(1)}(\mathbb F,S)
=(\mathfrak h^{(1)}(\mathbb F,S),\innerproduct)$. 
The pseudo $H$-type Lie algebra $\mathfrak H^{(1)}(\mathbb F,S)$ 
is called a pseudo $\Div H$-type Lie algebra of the first class. 
We denote the FGLA associated with $\mathfrak H^{(1)}(\mathbb F,S)$ 
by $\mathfrak h^{(1)}(\mathbb F,S)=\bigoplus\limits_{p=-1}^{-2}
\mathfrak h^{(1)}(\mathbb F,S)_p$. 

\begin{lemma}
Let $(r,s)$ be the signature of $S$. 
\begin{enumerate}
\item $\mathfrak H^{(1)}(\mathbb F,S)$ is  
isomorphic to 
$\mathfrak H^{(1)}(\mathbb F,1_{r,s})$. 
\item  $\mathfrak H^{(1)}(\mathbb F,1_{r,s})$ is equivalent to 
$\mathfrak H^{(1)}(\mathbb F,1_{s,r})$. 
\end{enumerate}
\label{lem31}
\end{lemma}
\begin{proof}
(1) 
There exists a real orthogonal matrix $P$ such that 
$PSP^{-1}=1_{r,s}$, where 
$1_{r,s}=
\begin{bmatrix} 1_r &O \\
O & -1_s \\
\end{bmatrix}$. 
We define a linear mapping $\varphi$ of 
$\mathfrak h^{(1)}(\mathbb F,1_{r,s})$ to 
$\mathfrak h^{(1)}(\mathbb F,S)$ as follows: 
$$\varphi(x)=
xP\quad (x\in \mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-1}), \quad 
\varphi(z)=z\quad (z\in \mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-2}).$$  
Then $\varphi$ is an isomorphism as a pseudo $H$-type Lie algebra. 
Hence  $\mathfrak H^{(1)}(\mathbb F,S)$ is  
isomorphic to 
$\mathfrak H^{(1)}(\mathbb F,1_{r,s})$. 

(2) We define a linear mapping $\psi$ of 
$\mathfrak h^{(1)}(\mathbb F,1_{r,s})$ to 
$\mathfrak h^{(1)}(\mathbb F,1_{s,r})$ as follows: 
$$\psi(x)=
xK_n \ (x\in \mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-1}), \quad 
\psi(z)=-z\ (z\in \mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-2}),$$ 
where  
$K_n$ is the $n\times n$ matrix whose $(i,j)$-component is 
$\delta_{i,n+1-j}$. 
Then $\psi$ is an isomorphism 
as a GLA. 
Moreover $\psi|\mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-2}$ is isometry and 
$\psi|\mathfrak h^{(1)}(\mathbb F,1_{r,s})_{-1}$ is anti-isometry. 
Hence $\mathfrak H^{(1)}(\mathbb F,1_{r,s})$ is equivalent to 
$\mathfrak H^{(1)}(\mathbb F,1_{s,r})$. 
\end{proof}
\begin{remark}
The $H$-type Lie algebra 
$\mathfrak H^{(1)}(\mathbb F,1_{r,s})$ coincides with 
$\mathfrak h'_{r,s}(\mathbb F)$ in 
\cite{KS16:0}. 
\label{rem31}
\end{remark}

\subsection{Pseudo $\Div H$-type Lie algebras of the second and the third classes}
\label{sec31}
Let $\mathbb F$ be $\mathbb C$, $\mathbb C'$, $\mathbb H$, $\mathbb H'$, 
$\mathbb O$ or $\mathbb O'$. 
We set 
$$\mathfrak g_{-1}=(\mathbb F(\gamma)^c)^n, \qquad 
\mathfrak g_{-2}=\mathbb F^c,$$ 
where we assume $n=1$ in case $\mathbb F=\mathbb O$ or $\mathbb O'$. 
Let $S$ be a real symmetric matrix of order $n$ such that $S^2=1_n$. 
We define a bracket operation $[\cdot\,,\,\cdot]$ on 
$\mathfrak m=\mathfrak g_{-2}\oplus \mathfrak g_{-1}$ as follows: 
$$[\alpha,\beta]=\pr_2(\alpha S\beta^*)\quad 
(\alpha,\beta\in \mathfrak g_{-1}),\quad 
[\mathfrak g_{-1},\mathfrak g_{-2}]=[\mathfrak g_{-2},\mathfrak g_{-2}]=0.$$ 
More explicitly, the bracket operation can be written as follows: 
if we put $\alpha=\alpha_1+\alpha_2\ell$ and $\beta=\beta_1+\beta_2\ell$ 
$(\alpha_1,\alpha_2,\beta_1,\beta_2\in(\mathbb F^c)^n$), then 
$$[\alpha,\beta]=\alpha_2S{}^t\beta_1-\beta_2S{}^t\alpha_1.$$ 
Then $\mathfrak m$ becomes a complex FGLA of the second kind. 
Moreover we define a symmetric bilinear form $\innerproduct$ on 
$\mathfrak m$ as follows:
$$\begin{aligned}
& \inph{\alpha}{\beta}=
R(\alpha S\beta^*)\quad   
(\alpha,\  \beta\in \mathfrak g_{-1}), \\
& \inph{z_1}{z_2}=-\gamma R(\overline{z_1}{z_2})\quad (z_1,z_2\in\mathfrak g_{-2}),\quad \inph{\mathfrak g_{-1}}{\mathfrak g_{-2}}=0. 
\end{aligned}$$
More explicitly, the bilinear form can be written as follows: 
if we put 
$\alpha=\alpha_1+\alpha_2\ell$ and $\beta=\beta_1+\beta_2\ell$ 
$(\alpha_1,\alpha_2,\beta_1,\beta_2\in(\mathbb F^c)^n$), 
then 
$$\inph{\alpha}{\beta}=R(\alpha_1S{}^t\overline{\beta_1}-
\gamma\overline{\beta_2}S{}^t\alpha_2).$$
For $z\in\mathfrak g_{-2}$ 
the linear mapping $J_z$ of $\mathfrak g_{-1}$ to itself defined by  
$$\inph{J_z(x)}{y}=\inph{z}{[x,y]}\qquad (x,y\in\mathfrak g_{-1})$$ 
satisfies 
$$J_z(\alpha)=-(z\ell)\alpha,\qquad 
J_z^2=\gamma\overline{z}z1_{\mathfrak g_{-1}}.$$ 
We denote by the same letter $\tau$ the conjugations of $\mathbb F^c$ and $\mathbb F(\gamma)^c$  
with respect to $\mathbb F$ and $\mathbb F(\gamma)$ respectively. 
We now extend $\tau$ to a grade-preserving involution of $\mathfrak m$ 
in a natural way, which is also denoted by the same letter. 
Next we define a grade-preserving involution $\kappa$ of $\mathfrak m$ as follows: 
$$\kappa(\alpha)=-\overline{\alpha_2}-\overline{\alpha_1}\ell,\qquad 
\kappa(z)=-\overline{z}, $$
where $\alpha=\alpha_1+\alpha_2\ell\in\mathfrak g_{-1}$ 
$(\alpha_1,\alpha_2\in(\mathbb F^c)^n$, 
$z\in\mathfrak g_{-2}$). 
We denote by $\mathfrak n^1$ and $\mathfrak n^2$ the sets of elements which are fixed under $\tau$ and  $\kappa\comp\tau$ respectively. 
Then $\mathfrak n^1$ and $\mathfrak n^2$ become graded subalgebras of 
$\mathfrak m_\mathbb R$ with 
$$\mathfrak n^i=\bigoplus_{p<0}\mathfrak n^i_p,\qquad 
\mathfrak n^i_p=\mathfrak n^i\cap\mathfrak g_p.$$
Explicitly the subspaces $\mathfrak n^i_p$ are described as follows: 
$$\begin{aligned}
&\mathfrak n^1_{-1}=\mathbb F(\gamma)^n, \qquad 
\mathfrak n^1_{-2}=\mathbb F, \\
& \mathfrak n^2_{-1}=\{\alpha_1+\hat{\tau}(\alpha_1)\ell:
\alpha_1\in(\mathbb F^c)^n\}, \qquad 
\mathfrak n^2_{-2}=\sqrt{-1}\mathbb R\oplus\ima(\mathbb F),
\end{aligned}$$
where $\hat{\tau}$ is a mapping of $\mathbb F^c$ to itself defined 
by $\hat{\tau}(x)=-\tau(\overline{x})$. 
We note that the bracket operation and the scalar product on $\mathfrak n^2$ can be written as follows: 
if we put $\alpha=\alpha_1+\hat{\tau}(\alpha_1)\ell$ and 
$\beta=\beta_1+\hat{\tau}(\beta_1)\ell$ 
$(\alpha_1,\beta_1\in(\mathbb F^c)^n$), then 
$$\begin{aligned}
&[\alpha,\beta]=\hat{\tau}(\alpha_1)S{}^t\beta_1
-\hat{\tau}(\beta_1)S{}^t\alpha_1, \\ 
& \langle\alpha\mid\beta\rangle=
R(\alpha_1S{}^t\overline{\beta_1}-\gamma\tau(\beta_1)S{}^t\tau(\overline{\alpha_1}))=(1-\gamma)R(\alpha_1S{}^t\overline{\beta_1}).
\end{aligned}$$ 
We always assume that $\gamma=-1$ when we consider $\mathfrak n^2$. 
Since $z\overline{z}\in\mathbb R$ for $z\in\mathfrak n^i_{-2}$ $(i=1,2)$, 
$\mathfrak n^1$ and $\mathfrak n^2$ are pseudo $H$-type Lie algebras. 
The pseudo $H$-type Lie algebra $(\mathfrak n^1,\innerproduct)$ 
is called a pseudo $\Div H$-type Lie algebra of the second class, 
which is denoted by $\mathfrak H^{(2)}(\mathbb F,S,\gamma)
=(\mathfrak h^{(2)}(\mathbb F,S,\gamma),\innerproduct)$. 
Also in case $\mathbb F=\mathbb H$, $\mathbb H'$, $\mathbb O$ or 
$\mathbb O'$, 
the pseudo $H$-type Lie algebra $(\mathfrak n^2,\innerproduct)$ 
is called a pseudo $\Div H$-type Lie algebra of the third class, 
which is denoted by $\mathfrak H^{(3)}(\mathbb F,S)
=(\mathfrak h^{(3)}(\mathbb F,S),\innerproduct)$. 
We denote the FGLA associated with 
$\mathfrak H^{(2)}(\mathbb F,S,\gamma)$ 
(resp. $\mathfrak H^{(3)}(\mathbb F,S)$) 
by $\mathfrak h^{(2)}(\mathbb F,S,\gamma)=\bigoplus\limits_{p=-1}^{-2}
\mathfrak h^{(2)}(\mathbb F,S,\gamma)_p$ 
(resp. 
$\mathfrak h^{(3)}(\mathbb F,S)=\bigoplus\limits_{p=-1}^{-2}
\mathfrak h^{(3)}(\mathbb F,S)_p)$. 
Note that $\mathfrak h^{(2)}(\mathbb C,S,\gamma)$ becomes 
a complex FGLA.  

\begin{lemma}
Let $(r,s)$ be the signature of $S$. 
\begin{enumerate}
\item 
$\mathfrak H^{(2)}(\mathbb F,S,\gamma)$ (resp. 
$\mathfrak H^{(3)}(\mathbb F,S)$)  
is isomorphic to 
$\mathfrak H^{(2)}(\mathbb F,1_{r,s},\gamma)$ 
(resp. $\mathfrak H^{(3)}(\mathbb F,1_{r,s})$). 
\item    
$\mathfrak h^{(2)}(\mathbb F,S,\gamma')$ is  
isomorphic to 
$\mathfrak h^{(2)}(\mathbb F,1_{r+s},\gamma)$ 
as a GLA. 
\item $\mathfrak H^{(2)}(\mathbb F,1_{r,s})$  
is equivalent to 
$\mathfrak H^{(2)}(\mathbb F,1_{s,r})$.   
\item When $\mathbb F=\mathbb H$ or $\mathbb H'$, 
$\mathfrak H^{(3)}(\mathbb F,1_{r,s})$ is isomorphic to 
$\mathfrak H^{(3)}(\mathbb F,1_{r+s})$. 
Consequently, for a fixed $\mathbb F$ 
the $\mathfrak H^{(3)}(\mathbb F,S)$ are mutually isomorphic. 
\end{enumerate}
\label{lem32}
\end{lemma}
\begin{proof}
As in Lemma \ref{lem31} we can prove (1) and (3).  

(2) There exists a real orthogonal matrix $P$ such that 
$PSP^{-1}=1_{r,s}$. We define a linear mapping of 
$\mathfrak h^{(2)}(\mathbb F,1_{r+s},\gamma')$ to 
$\mathfrak h^{(2)}(\mathbb F,S,\gamma)$ as follows: 
$$\varphi(\alpha_1+\alpha_2\ell)=
\alpha_1P+\alpha_21_{r,s}P\ell\quad(\alpha_1,\alpha_2\in \mathbb F^n), 
\qquad \varphi(z)=z
\quad(z\in\mathfrak h^{(2)}(\mathbb F,1_{r+s},\gamma')_{-2}).$$ 
Then $\varphi$ is an isomorphism 
as a GLA. 

(4) First we assume that $\mathbb F=\mathbb H'$. 
We define 
a linear mapping of 
$\mathfrak h^{(3)}(\mathbb F,1_{r+s})$ to 
$\mathfrak h^{(3)}(\mathbb F,1_{r,s})$ as follows: 
$$\varphi(\alpha_1+\hat{\tau}(\alpha_1)\ell)=
\eta(\alpha_1)Q+\hat{\tau}(\eta(\alpha_1)Q)\ell
\quad (\alpha_1\in (\mathbb F^c)^n), 
\quad \varphi(z)=z\quad (z\in\mathfrak h^{(3)}(\mathbb F,1_{r+s})_{-2}).$$ 
Here $Q=
\begin{bmatrix}
1_r & O \\
O & \ell_0 1_s \\
\end{bmatrix}$ and 
$\eta$ is the mapping of $(\mathbb F^c)^n$ to itself defined by 
$\eta(\alpha_r,\alpha_s)=
(\alpha_r,\overline{\alpha_s})$ $(\alpha_r\in(\mathbb F^c)^r,
 \alpha_s\in(\mathbb F^c)^s)$. 
Then $\varphi$ is an isomorphism of 
$\mathfrak H^{(3)}(\mathbb F,1_{r+s})$ onto 
$\mathfrak H^{(3)}(\mathbb F,1_{r,s})$. 

Next we assume that $\mathbb F=\mathbb H$. 
We define 
a linear mapping of 
$\mathfrak h^{(3)}(\mathbb F,1_{r+s})$ to 
$\mathfrak h^{(3)}(\mathbb F,1_{r,s})$ as follows: 
$$\varphi(\alpha_1+\hat{\tau}(\alpha_1)\ell)=
\eta(\alpha_1)R+\hat{\tau}(\eta(\alpha_1)R)\ell
\ (\alpha_1\in (\mathbb F^c)^n), 
\quad \varphi(z)=z\ (z\in\mathfrak h^{(3)}(\mathbb F,1_{r+s})_{-2}),$$ 
where $R=
\begin{bmatrix}
1_r & O \\
O & \sqrt{-1}\ell_0 1_s \\
\end{bmatrix}$. 
Then $\varphi$ is an isomorphism of 
$\mathfrak H^{(3)}(\mathbb F,1_{r,s})$ onto 
$\mathfrak H^{(3)}(\mathbb F,1_{r+s})$. 
\end{proof}
\begin{remark}
The $H$-type Lie algebra 
$\mathfrak H^{(2)}(\mathbb F,1_{r+s},-1)$ coincides with 
$\mathfrak h_{r+s}(\mathbb F)$ in 
\cite{KS16:0}. 
\label{rem32}
\end{remark}
\subsection{Pseudo $\Div H$-type Lie algebras with $\dim\mathfrak n_{-2}=1$}
 (cf. \cite[Proposition 4.5]{AS14:0}). 
Now let $(\mathfrak n,\innerproduct)$ be a pseudo $\Div H$-type Lie algebra with $\dim \mathfrak n_{-2}=1$, 
that is, $(\mathfrak n,\innerproduct)$ is  
$\mathfrak H^{(1)}(\mathbb C,S)$ or 
$\mathfrak H^{(1)}(\mathbb C',S)$.  
Note that $\mathfrak h^{(1)}(\mathbb C,S)$ is isomorphic to 
$\mathfrak h^{(1)}(\mathbb C',S)$ as a GLA.  
Since $\dim \mathfrak n_{-2}=1$ and the FGLA $\mathfrak n$ 
is non-degenerate, the prolongation of $\mathfrak n$ is isomorphic to 
a real contact algebra $K(N/2,\mathbb R)$, where $N=\dim \mathfrak n_{-1}$. 
(For the details of contact algebras, see \cite{Kac68:01}). 
By definition an SGLA $\gla l$ is is said to be of contact type 
if the negative part is an FGLA of the second kind and 
$\dim \mathfrak l_{-2}=1$. 
The negative part of a finite dimensional SGLA $\gla l$ of 
contact type is uniquely determined by $\dim \mathfrak l_{-1}$ 
up to isomorphism. 
A finite dimensional real SGLA $\gla l$ of contact type has the negative part isomorphic to 
$\mathfrak h^{(1)}(\mathbb C,S)$ and is one of the following types: 
$$\begin{aligned}
& 
(({\rm AI})_l,\{\alpha_1,\alpha_l\}), \ 
(\textrm{(AIIIa)}_{l,p},\{\alpha_1,\alpha_l\}), \   
(\textrm{(AIIIb)}_{l},\{\alpha_1,\alpha_l\}), \   
(\textrm{(AIV)}_{l},\{\alpha_1,\alpha_l\}), \  
(\textrm{(BI)}_{l},\{\alpha_2\}), \\  
&(\textrm{(CI)}_{l},\{\alpha_1\}), \ 
(\textrm{(DI)}_{l},\{\alpha_2\}), \ 
(\textrm{EI},\{\alpha_2\}), \  
(\textrm{EII},\{\alpha_2\}), \ 
(\textrm{EIII},\{\alpha_2\}), \ 
(\textrm{EIV},\{\alpha_2\}), \\  
&(\textrm{EV},\{\alpha_1\}), \ 
 (\textrm{EVI},\{\alpha_1\}), \  
(\textrm{EVII},\{\alpha_1\}), \  
(\textrm{EVIII},\{\alpha_8\}), \  
(\textrm{EIX},\{\alpha_8\}), \  
(\textrm{FI},\{\alpha_1\}), \  
(\textrm{G},\{\alpha_2\}),  
\end{aligned}$$
For the description of finite dimensional SGLAs, we use the notations in \cite[\S3]{yam93:1}. 

\subsection{Pseudo $\Div H$-type Lie algebras with $\dim\mathfrak n_{-2}=2$} 
(cf. \cite[Proposition 4.4]{AS14:0}). 
Now let $(\mathfrak n,\innerproduct)$ be a pseudo $\Div H$-type Lie algebra 
with $\dim \mathfrak n_{-2}=2$, 
that is, $(\mathfrak n,\innerproduct)$ is  
$\mathfrak H^{(2)}(\mathbb F,S,\gamma)$ 
$(\mathbb F=\mathbb C$ or $\mathbb C')$.  
We define an endomorphism $I$ of 
$\mathfrak n$ as follows: 
$$I(\alpha)=-\gamma J_1J_{\ell_0}(\alpha)=\ell_0(\alpha),\quad  
I(z)= \ell_0z\quad \text{if}\ (\mathfrak n,\innerproduct)=\mathfrak H^{(2)}(\mathbb F,S,\gamma)
$$
then $I$ satisfies 
$I^2=\gamma_0 1_{\mathfrak n}$, 
$[Ix,y]=I[x,y]$, and $\inph{Ix}{y}+\inph{x}{Iy}=0$. 

(i) Firstly we assume $(\mathfrak n,\innerproduct)
=\mathfrak H^{(2)}(\mathbb C,S,\gamma)$; 
then $(\mathfrak n,I)$ becomes a complex Lie algebra. 
The prolongation of the  complex FGLA $\mathfrak n$ is isomorphic to a complex contact algebra $K(N/4;\mathbb C)$, where $N=\dim \mathfrak n_{-1}$. 
Hence 
the prolongation of the real FGLA $\mathfrak n$ is isomorphic to 
$K(N/4;\mathbb C)_\mathbb R$ of a complex contact algebra $K(N/4;\mathbb C)$. 
The signature of $\innerproduct_{-2}$ is $(2,0)$ 
(resp. $(0,2)$). 
The negative part of a finite dimensional complex SGLA $\gla l$ of 
contact type has the negative part isomorphic to 
$\mathfrak h^{(2)}(\mathbb C,S,\gamma)$ and is one of the following types: 
$$\begin{aligned}
& (A_l,\{\alpha_1,\alpha_l\}), \ 
(B_{l},\{\alpha_2\}), \ 
(C_l,\{\alpha_1\}), \ 
(D_{l},\{\alpha_2\}), \\
& (E_6,\{\alpha_2\}), \ 
(E_7,\{\alpha_1\}), \ 
(E_8,\{\alpha_8\}), \ 
(F_4,\{\alpha_1\}), \ 
(G_2,\{\alpha_2\}), 
\end{aligned}$$

(ii) Next we assume 
$(\mathfrak n,\innerproduct)
=\mathfrak H^{(2)}(\mathbb C',S',\gamma)$. 
We set $\mathfrak n^\pm=\{\alpha\in \mathfrak n:I(\alpha)=\pm\alpha\}$ and $(\mathfrak n^\pm)_p=\mathfrak n_p\cap\mathfrak n^\pm$; 
then $\mathfrak n^+$ and $\mathfrak n^-$ are ideals of $\mathfrak n$ 
such that $\mathfrak n=\mathfrak n^+\oplus\mathfrak n^-$, 
$[\mathfrak n^+,\mathfrak n^-]=0$, 
$\inph{\mathfrak n^+}{\mathfrak n^+}
=\inph{\mathfrak n^-}{\mathfrak n^-}=0$. 
Let $\check{\mathfrak g}^+=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^+$ and $\check{\mathfrak g}^-
=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^-$ 
be the prolongation of $\mathfrak n^+$  and $\mathfrak n^-$ respectively. 
$\check{\mathfrak g}^+=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^+$ and $\check{\mathfrak g}^-
=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^-$ 
are both isomorphic to a real contact algebra $K(N/4;\mathbb R)$.  
Hence the prolongation $\cgla g$ of 
the FGLA $\mathfrak n$ is 
the direct sum of $\check{\mathfrak g}^+=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^+$ and $\check{\mathfrak g}^-
=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^-$ and hence is 
isomorphic to $K(N/4;\mathbb R)\oplus K(N/4;\mathbb R)$. 
Let $\gla g$ be the prolongation of $(\mathfrak n,[\innerproduct_{-1}])$; 
then $\mathfrak g_0=\mathbb RE_+\oplus\mathbb RE_-\oplus\mathfrak a$, where 
$\mathfrak a=\{~D-D^\top:D\in\check{\mathfrak g}^+_0, [D,\mathfrak n_{-2}]=0~\}$, 
where $E_+$ (resp. $E_-$) is the characteristic element of 
$\check{\mathfrak g}^+=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^+$ 
(resp. $\check{\mathfrak g}^-
=\bigoplus\limits_{p\in\mathbb Z}\check{\mathfrak g}_p^-$) and $D^\top$ is the adjoint of $D$ with respect to $\innerproduct$. 
The ideal $\mathfrak a$ of $\check{\mathfrak g}_0$ is isomorphic to 
$\mathfrak{sp}(\mathfrak n^+_{-1})$. 
Therefore the $\mathfrak g_0$-module $\mathfrak g_{-1}$ is completely reducible. From these results, we can easily prove that $\mathfrak g_2=0$. 

\subsection{Matricial models of pseudo $\Div H$-type Lie algebras of the first class}
Let $\mathbb F$ be $\mathbb C$, $\mathbb H$, 
$\mathbb C'$ or $\mathbb H'$. 
We put 
$\mathfrak l=
\mathfrak{sl}(n+2,\mathbb F)$ $(n\geqq1)$; 
then $\mathfrak l$ is a real semisimple Lie algebra. 
We define an $n\times n$ symmetric real matrix $S_{p,q}$ as follows: 
$$S_{p,q}=
\begin{bmatrix}
0 & 0 & K_p \\
0 & 1_q & 0 \\
K_p & 0 & 0 \\
\end{bmatrix}\qquad (p\geqq1,q\geqq0,2p+q=n+2\geqq3).$$
Here the center column and the center row of $S_{p,q}$ should be deleted 
when $q=0$. 
Then $S_{p,q}$ is a symmetric real matrix with signature $(p+q,p)$. 
We put 
$
\mathfrak{g}=\{\,X\in\mathfrak l:
X^*S_{p,q}+S_{p,q}X=O\,\}$; then 
$$\mathfrak{g}=\left\{
X=\begin{bmatrix}
X_{11} & X_{12} & X_{13} \\
X_{21} & X_{22} & -S_{p-1,q}X^*_{12} \\
X_{31} & -X^*_{21}S_{p-1,q} & -\overline{X_{11}} \\
\end{bmatrix}\in\mathfrak l
:
\begin{aligned} 
& X_{11}\in \mathbb F,\ X_{12}\in M(1,n,\mathbb F), \\ 
& X_{21}\in M(n,1,\mathbb F), \\ 
&X_{31},X_{13}\in \ima \mathbb F, 
X_{22}\in \mathfrak{gl}(n',\mathbb F), \\
&X_{22}+S_{p-1,q}X^*_{22}S_{p-1,q}=O
\end{aligned}\right\},
$$
where we set $S_{0,m}=1_m$. 
Here $M(p,q,\mathbb F)$ denotes the set of $\mathbb F$-valued $p\times q$-matrices. 
We define subspaces $\mathfrak g_p$ of $\mathfrak g$ as follows: 
$$\begin{aligned}
&\mathfrak g_{-2}
 =\left\{\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
x_{31} & 0 & 0 \\
\end{bmatrix}\in\mathfrak g:
x_{31}\in\ima \mathbb F
\right\}, \\
& 
\mathfrak g_{-1}=\left\{\begin{bmatrix}
0 & 0 & 0 \\
x_{21} & 0 & 0 \\
0 & -x^*_{21}S_{p-1,q} & 0 \\
\end{bmatrix}\in\mathfrak g: 
x_{21}\in M(n,1,\mathbb F)\right\}, \\
& 
\mathfrak g_{0}=\left\{\begin{bmatrix}
x_{11} & 0 & 0 \\
0 & x_{22} & 0 \\
0 & 0 & -\overline{x_{11}} \\
\end{bmatrix}\in\mathfrak g: 
\begin{aligned} 
& x_{11}\in \mathbb F, 
x_{22}\in \mathfrak{gl}(n,\mathbb F), \\
& x_{22}+S_{p-1,q}x^*_{22}S_{p-1,q}=O
\end{aligned}
\right\}, \\
& 
\mathfrak g_p=\{~X\in\mathfrak g:
{}^tX\in\mathfrak g_{-p}~\}\quad (p=1,2), 
\quad \mathfrak g_p=\{0\} \quad (|p|>2). 
\\
\end{aligned}$$
Then $\gla g$ becomes a GLA whose negative part $\mathfrak m$ is an FGLA of the second kind. 
We define a linear mapping of $\mathfrak h^{(1)}(\mathbb F,S_{p-1,q})$ into 
$\mathfrak g_-$ as follows: 
$$\varphi(x)=
\begin{bmatrix}
0 & 0 & 0 \\
x & 0 & 0 \\
0 & -x^*S_{p-1,q} & 0 \\
\end{bmatrix}
\quad (x\in \mathbb F^{p+q-1}),\quad \varphi(z)=\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
z & 0 & 0 \\
\end{bmatrix}\quad (z\in\mathfrak n_{-2});
$$
then $\varphi$ becomes a GLA isomorphism. 
We define a symmetric bilinear form $\innerproduct $ on $\mathfrak g_{-}$ as follows:  
$$\begin{aligned}
\inph{X}{Y}&=2\rea\tr(XSY^*) \quad (X,Y\in\mathfrak g_{-1}),\quad 
\inph{X}{Y}=\rea\tr(XY^*) \quad (X,Y\in\mathfrak g_{-2}),\\ 
\inph{X}{Y}&=0\quad (X\in\mathfrak g_{-2},Y\in\mathfrak g_{-1})
\end{aligned}$$
Then $(\mathfrak g_{-},\innerproduct)$ becomes a pseudo $H$-type Lie algebra 
and $\varphi$ is isomorphism of $\mathfrak H^{(1)}(\mathbb F,S_{p-1,q})$ 
onto $(\mathfrak g_{-},\innerproduct)$. 
Since 
$\ad(\mathfrak g_0)|\mathfrak g_{-1}\subset\mathfrak{co}(\mathfrak g_{-1},g)$, 
$\gla g$ is the prolongation of $(\mathfrak g_-,[\innerproduct_{-1}])$. 
From these results, \cite[Theorem 3.6]{AS14:0}, \cite[\S3]{gom96:0} and \cite{yat18:01}, a finite dimensional real SGLA $\gla s$ that is isomorphic to 
the prolongation of the cps-FGLA $(\mathfrak n,[\innerproduct_{-1}])$ associated with 
a pseudo $\Div H$-type Lie algebras $(\mathfrak n,\innerproduct)$ 
of the first class is one of the following:   

\medskip
\begin{tabular}{|c|c|c|p{9.5cm}|}\hline
$\mathbb F$ & $\sgn \innerproduct_{-2}$ & $\mathfrak s$ & 
the gradation of $\mathfrak s$ \\ \hline\hline 
$\mathbb C$ & $(1,0)$ & $\mathfrak{su}(p+q,p) $ & 
$(\textrm{(AIIIa)}_{l,p},\{\alpha_1,\alpha_l\})$ 
{\tiny $(l=n-1=2p+q-1, p\geqq2,q\geqq1)$}, 
$(\textrm{(AIIIb)}_{l},\{\alpha_1,\alpha_l\})$ 
{\tiny $(l=n-1=2p-1, p\geqq2,q=0)$}, 
$(\textrm{(AIV)}_{l},\{\alpha_1,\alpha_l\})$ 
{\tiny $(l=n-1=q+1,p=1,q\geqq1)$} 
 \\ \hline
$\mathbb C'$ & $(0,1)$ & $\mathfrak{sl}(2p+q,\mathbb R)$ & $(\textrm{(AI)}_l,\{\alpha_1,\alpha_l\})$ \\ \hline
$\mathbb H$ &  $(3,0)$ & $\mathfrak{sp}(p+q,p)$ & $(\textrm{(CIIa)}_{l,p},\{\alpha_2\})$ 
{\tiny $(l=n=2p+q\geqq3,p,q\geqq1)$}, 
$(\textrm{(CIIb)}_{l},\{\alpha_2\})$ 
{\tiny ($n=l=2p\geqq3,q=0$)} 
  \\ \hline
$\mathbb H'$ & $(1,2)$ & $\mathfrak{sp}(2p+q,\mathbb R)$& $(\textrm{(CI)}_l,\{\alpha_2\})$ 
{\tiny $(l=n=2p+q\geqq3)$} \\ \hline
$\mathbb O$ & $(7,0)$ & {\rm FII} & $({\rm FII},\{\alpha_4\})$ \\ \hline
$\mathbb O'$ &  $(3,4)$ & {\rm FI} & $({\rm FI},\{\alpha_4\})$ \\ \hline
\end{tabular}

\medskip\noindent
In particular, if $\dim\mathfrak s_{-2}\geqq3$, then 
$\gla s$ is the prolongation of $\mathfrak s_-$. 

\subsection{Matricial Models of pseudo $\Div H$-type Lie algebras of the second class} 
Let $\mathbb F=\mathbb C, \mathbb C',\mathbb H,\mathbb H'$. 
Let $\gla g$ be a finite dimensional semisimple GLA $\mathfrak{sl}(n+2,\mathbb F)$ with the the following gradation $(\mathfrak g_p)$. 

$$\begin{aligned}
&\mathfrak g_{-2}
 =\left\{\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
x_{31} & 0 & 0 \\
\end{bmatrix}\in\mathfrak g:
x_{31}\in \mathbb F\right\}, \\
&\mathfrak g_{-1}
 =\left\{\begin{bmatrix}
0 & 0 & 0   \\
x_{21} & 0 & 0 \\
0& x_{32} & 0  \\
\end{bmatrix}\in\mathfrak g:
x_{21}\in M(n,1,\mathbb F), x_{32}\in M(1,n;\mathbb F)
\right\}, \\
\end{aligned}$$
Note that $\gla g$ is an SGLA except for the case $\mathbb F=\mathbb C'$. 
We consider an FGLA $\mathfrak H^{(2)}(\mathbb F,S,\gamma)$.
That is, 
$$\mathfrak h^{(2)}(\mathbb F,S,\gamma)_{-1}=\mathbb F(\gamma)^n,\quad
\mathfrak h^{(2)}(\mathbb F,S,\gamma)_{-2}=\mathbb F,$$
where $S$ is a real symmetric matrix of order $n$ such that $S^2=1_n$. 
We define a linear mapping $\varphi$ of $\mathfrak h^{(2)}(\mathbb F,S,\gamma)$ to $\mathfrak g_-$ 
as follows:
$$\varphi(\alpha_1+\alpha_2\ell)=
\begin{bmatrix}
0 & 0 & 0  \\
{}^t\alpha_1 & 0 & 0   \\
0& \alpha_2S & 0  \\
\end{bmatrix},\qquad 
\varphi(z)=
\begin{bmatrix}
0 & 0 & 0  \\
0 & 0 & 0  \\
z & 0 & 0   \\
\end{bmatrix}.$$
Then $\varphi$ is a GLA isomorphism. 
Moreover we define a non-degenerate symmetric bilinear form 
on $\mathfrak g_-$ as follows: 
$$\begin{aligned}
& \inph{X}{Y}=\rea(\transpose{x_{21}}S\overline{y_{21}}
-\gamma x_{32}Sy_{32}^*)
, \\
& \inph{Z}{W}=-\gamma\rea(z_{31}\overline{w_{31}})
\quad (Z,W\in\mathfrak g_{-2}),\quad \inph{\mathfrak g_{-1}}{\mathfrak g_{-2}}=0,  
\end{aligned}$$
The negative part of $\gla g$ equipped with this scalar product becomes 
a pseudo $H$-type Lie algebra which is isomorphic to 
$\mathfrak H^{(2)}(\mathbb F,S,\gamma)$ as a pseudo $H$-type Lie algebra. 

\begin{enumerate}
\renewcommand{\labelenumi}{\textbf{Case \arabic{enumi}:}}
\item  
$\mathbb F=\mathbb C$. $\mathfrak g$ is equal to $\mathfrak{sl}(n+2,\mathbb C)_\mathbb R$. Hence the GLA $\gla g$ is a finite dimensional SGLA of type $(A_{l},\{\alpha_1,\alpha_l\})$ $(l=n+1)$. 
If $\gamma=-1$ (resp. $\gamma=1$), then the signature of $\inph{\cdot}{\cdot}_{-2}$ is $(2,0)$ (resp. $(0,2)$). 
\item 
$\mathbb F=\mathbb C'$. Since $\mathbb C'$ is isomorphic to $\mathbb R\oplus 
\mathbb R$ as a $\mathbb R$-algebra, 
$\mathfrak g$ is isomorphic to 
$\mathfrak{sl}(n+2,\mathbb R)\times \mathfrak{sl}(n+2,\mathbb R)$. 
Hence the GLA $\gla g$ is a semisimple GLA of type 
$(({\rm AI})_{l},\{\alpha_1,\alpha_{l}\})\times( ({\rm AI})_l,\{\alpha_1,\alpha_{l}\})$, where $l=n+1$. 
The signature of $\inph{\cdot}{\cdot}_{-2}$ is $(1,1)$. 
\item
$\mathbb F=\mathbb H$. 
The GLA $\gla g$ is a finite dimensional SGLA of type $(({\rm AII})_{l},\{\alpha_2,\alpha_{l-1}\})$, 
where $l=2n+1$. 
If $\gamma=-1$ (resp. $\gamma=1$), then the signature of $\inph{\cdot}{\cdot}_{-2}$ is $(4,0)$ (resp. $(0,4)$). 
\item 
$\mathbb F=\mathbb H'$. 
Since $\mathbb H'$ is isomorphic to $M_2(\mathbb R)$ as a $\mathbb R$-algebra, 
$\mathfrak g$ is isomorphic to $\mathfrak{sl}(2n+2,\mathbb R)$. 
Hence the GLA $\gla g$ is a finite dimensional SGLA of type $(({\rm AI})_{l},\{\alpha_2,\alpha_{l-1}\})$, where $l=2n-1$. 
The signature of $\inph{\cdot}{\cdot}_{-2}$ is $(2,2)$. 
\end{enumerate}
From these results, \cite[Theorem 3.6]{AS14:0} and \cite[\S3]{gom96:0}, 
a finite dimensional real SGLA $\gla s$ with $\dim\mathfrak s_{-2}\geqq3$ 
whose negative part is isomorphic to a pseudo $\Div H$-type Lie algebra of the second 
class is the prolongation of $\mathfrak s_-$ and is one of the following: 

\medskip
\begin{tabular}{|c|c|c|c|c|}\hline
$\mathbb F$ & $\gamma$ & $\sgn \innerproduct_{-2}$ & $\mathfrak s$ & the gradation \\ \hline\hline
$\mathbb H$ &  $-1$ & $(4,0)$ & $\mathfrak{sl}(m,\mathbb H)$ & 
$({\rm (AII)}_l,\{\alpha_2,\alpha_{l-1}\})$ \\ \hline
$\mathbb H$ &  $1$ & $(0,4)$ & $\mathfrak{sl}(m,\mathbb H)$ & 
$({\rm (AII)}_l,\{\alpha_2,\alpha_{l-1}\})$ \\ \hline
$\mathbb H'$ & $-1$  & $(2,2)$ & $\mathfrak{sl}(m,\mathbb R)$ & 
$({\rm (AI)}_l,\{\alpha_2,\alpha_{l-1}\})$ \\ \hline
$\mathbb O$  & $-1$ & $(8,0)$ & ${\rm EIV}$ & $({\rm EIV},\{\alpha_1,\alpha_6\})$ \\ \hline
$\mathbb O$  & $1$ & $(0,8)$ & ${\rm EIV}$ & $({\rm EIV},\{\alpha_1,\alpha_6\})$
\\ \hline
$\mathbb O'$ &  $-1$ & $(4,4)$ & ${\rm EI}$& $({\rm EI},\{\alpha_1,\alpha_6\})$ \\ \hline
\end{tabular}

\subsection{Matricial models of pseudo $\Div H$-type Lie algebras of the third class}
Let $\mathfrak g$ be the simple Lie algebra $\mathfrak{su}(p+q,p)$. 
We define subspaces $\mathfrak g_p$ of $\mathfrak g$ as follows: 
$$\begin{aligned}
&\mathfrak g_{-2}
 =\left\{\begin{bmatrix}
0 & 0 & 0  & 0 &0 \\
0 & 0 & 0  & 0 &0 \\
0 & 0 & 0  & 0 &0 \\
z_{41} & z_{42} & 0  & 0 &0 \\
z_{51} & -\overline{z_{41}} & 0  & 0 &0 \\
\end{bmatrix}\in\mathfrak g:
z_{41}\in\mathbb K,
z_{42},z_{51}\in\sqrt{-1}\mathbb R
\right\}, \\
&\mathfrak g_{-1}
=\left\{\begin{bmatrix}
0 & 0 & 0  & 0 &0 \\
x_{31} & x_{32} & 0  & 0 &0 \\
0 & 0 & -x_{32}^*S_{p-2,q}  & 0 &0 \\
0 & 0 & -x_{31}^*S_{p-2,q}  & 0 &0 \\
\end{bmatrix}\in\mathfrak g:
x_{31},x_{32}\in M(2p+q-4,1)\right\}, \\
& 
\mathfrak g_{0}=\left\{\begin{bmatrix}
X_{11} & 0 & 0 \\
0 & X_{22} & 0 \\
0 & 0 & -\overline{X_{11}} \\
\end{bmatrix}\in\mathfrak g: 
\begin{aligned} 
& X_{11}\in M(2,2), 
X_{22}\in \mathfrak{gl}(n',\mathbb K), \\
& X_{22}+S_{p-2,q}X^*_{22}S_{p-2,q}=O
\end{aligned}
\right\}, \\
& 
\mathfrak g_p=\{~X\in\mathfrak g:
{}^tX\in\mathfrak g_{-p}~\}\quad (p=1,2), 
\quad \mathfrak g_p=\{0\} \quad (|p|>2). 
\\
\end{aligned}$$
For convenience, we denote by $X=(x_{31},x_{32})$ and 
$Z=(z_{41},z_{42},z_{51})$ elements 
$$
X =
\begin{bmatrix}
0 & 0 & 0  & 0 &0 \\
0 & 0 & 0  & 0 &0 \\
x_{31} & x_{32} & 0  & 0 &0 \\
0 & 0 & -x_{32}^*S_{p-2,q}  & 0 &0 \\
0 & 0 & -x_{31}^*S_{p-2,q}  & 0 &0 \\
\end{bmatrix},
\quad Z =\begin{bmatrix}
0 & 0 & 0 & 0 & 0   \\
0 & 0 & 0 & 0 & 0   \\
0 & 0 & 0 & 0 & 0   \\
z_{41} & z_{42} & 0 & 0 & 0   \\
z_{51} & -\overline{z_{41}} & 0 & 0 & 0  \\
\end{bmatrix}$$ 
of $\mathfrak g_{-1}$ and $\mathfrak g_{-2}$ respectively. 
Then $\gla g$ becomes a GLA whose negative part $\mathfrak m$ is an FGLA of the second kind. 
For $X=(x_{31},x_{32}),Y=(y_{31},y_{32})\in\mathfrak g_{-1}$ 
$$[X,Y]=
( -x_{32}^*S'y_{31}+ y_{32}^*S'x_{31}, -x_{32}^*S'y_{32}+ y_{32}^*S'x_{32},
-x_{31}^*S'y_{31}+ y_{31}^*S'x_{31}),$$
where $S'=S_{p-2,q}$. 
For $X=(x_{31},x_{32})\in\mathfrak g_{-1}$ 
we denote by $X_{31}$ the $(2p+q-4)\times 2$ submatrix 
$\begin{bmatrix}
x_{31} & x_{32}\\
\end{bmatrix}$ of $X$. 
Also we use the notation 
$x_{3i}=\begin{bmatrix}
x_{3i}^{(1)} \\ x_{3i}^{(2)} \\ x_{3i}^{(3)}
\end{bmatrix}$, 
where 
$x_{3i}^{(1)}$ and $x_{3i}^{(3)}$ are $(p-2)\times 1$ matrices and 
$x_{3i}^{(2)}$ is a $q\times 1$ matrix. 
We define a non-degenerate symmetric bilinear form $\inph{\cdot}{\cdot}$ 
on 
$\mathfrak m$ as follows: 
$$\begin{aligned}
\inph{X}{Y}& =
\rea(\tr(Q_1\transpose{X_{31}}Q_{p+m}Y_{31}))\\
\inph{Z}{W} &=
\frac{\zeta_0}2(\det(Z_{31}+W_{31})-\det(Z_{31})-\det(W_{31})) \\
&=\frac{\zeta_0}2(-\alpha_1\overline{\alpha_2}-\alpha_2\overline{\alpha_1}-\beta_1\gamma_2-\beta_2\gamma_1), \qquad 
\inph{\mathfrak g_{-1}}{\mathfrak g_{-2}}=0, 
\end{aligned}
$$
where $m=q/2$, 
$Q_m=
\begin{bmatrix}
O  & K_{m} \\ \\
-K_{m} & O  \\
\end{bmatrix}
$ and $\zeta_0=\pm1$. 
For $Z\in\mathfrak g_{-2}$ let $J_Z$ be the mapping 
of $\mathfrak g_{-1}$ to itself defined by 
$$\inph{J_Z(X)}{Y}=\inph{Z}{[X,Y]}\qquad (X,Y\in\mathfrak g_{-1}).$$ 
Then 
$$
J_Z(X)_{31}=P_{p,q}\overline{X_{31}}PZ,
$$
where 
$P_{p,q}=\begin{bmatrix}
E_{p-2} & O & O \\
O & Q_m & O \\
O & O & -E_{p-2} \\
\end{bmatrix}$. 
Furthermore we obtain that 
$$  
J_Z^2(X)_{31}=\zeta_0P_{p,q}^2\overline{\overline{X_{31}}PZ}PZ=
-\inph{Z}{Z}\zeta_0
\begin{bmatrix} 
1_{p-2} & O & O \\
O & -1_q & O \\
O & O & 1_{p-2} \\
\end{bmatrix}X_{31}.$$ 

\subsubsection{Case of signature $(1,3)$}
We assume that $p\geqq3$, $q=0$ and $\zeta_0=1$. 
Then $(\mathfrak g_-,\innerproduct)$ becomes a pseudo $H$-type 
Lie algebra. This result is a little generalization of 
\cite[Theorem 8]{FMM17:0}. 
Note that the signature of the restriction of $\innerproduct$ to 
$\mathfrak g_{-2}$ is $(1,3)$ and $\gla g$ 
is a finite dimensional SGLA of type $({\rm (AIIIb)}_{l},\{\alpha_2,\alpha_{l-1}\})$, where $l=2p-1$. 
We define a linear mapping $\varPsi$ of $\mathfrak g_-$ to 
$\mathfrak H^{(3)}(\mathbb H',K_{p-2})$ as follows: 
$$\begin{aligned}
\varPsi(X)&=
\alpha_1+\hat{\tau}(\alpha_1)\ell, \\
\alpha_1 &=\frac12[
(-\Re(x_{31}^{(1)}-x_{32}^{(3)})+
i\Im(x_{31}^{(3)}-x_{32}^{(1)}))
+(\Im(x_{31}^{(3)}+x_{32}^{(1)})
+i\Re(x_{31}^{(1)}+x_{32}^{(3)}))\ell_0) \\
& 
+\sqrt{-1}((\Im(x_{31}^{(1)}-x_{32}^{(3)})+
i\Re(x_{31}^{(3)}-x_{32}^{(1)}))
+(\Re(x_{31}^{(3)}+x_{32}^{(1)})
-i\Im(x_{31}^{(1)}+x_{32}^{(3)})
\ell_0))], \\
\varPsi(Z)&=
\sqrt{-1}\Im(\alpha)-\frac{\Im(\beta+\gamma)}{2}i
+\frac{\Im(\beta-\gamma)}{2}\ell_0
+\Re(\alpha)i\ell_0, 
\end{aligned}$$
where $X=(x_{31},x_{32})\in\mathfrak g_{-1}$ and 
$Z=(\alpha, \beta,\gamma) \in\mathfrak g_{-2}$.
Here for a complex number $z=a+bi$ ($a,b\in \mathbb R$) 
we denote the real part $a$ (resp. the imaginary part $b$) of $z$ 
by $\Re(z)$ (resp. $\Im(z)$). 
$\varPsi$ is isomorphic to $\mathfrak g_-$ onto 
$\mathfrak n$ as a pseudo $H$ type Lie algebra. 

\subsubsection{Case of signature $(3,1)$}
We assume that $p=2$, $q=2m$, $m\geqq1$ and $\zeta_0=-1$. 
Note that the signature of the restriction of $\innerproduct$ to 
$\mathfrak g_{-2}$ is $(3,1)$ and 
$\gla g$ is a finite dimensional SGLA of type $({\rm (AIIIa)}_{l,2},\{\alpha_2,\alpha_{l-1}\})$, where $l=2m+3$. 
We define a linear mapping $\varPsi$ of $\mathfrak g_-$ to $\mathfrak H^{(3)}(\mathbb H,K_{q/2})$ as follows: 
$$\begin{aligned}
\varPsi(X)&=
\alpha_1+\hat{\tau}(\alpha_1)\ell, \\
\alpha_1&=
\frac12[
(\Re(x_{31}^{1}-x_{32}^{2})+
i\Im(x_{31}^{2}-x_{32}^{1}))
+(\Re(x_{31}^{2}+x_{32}^{1})
+i\Im(x_{31}^{1}+x_{32}^{2}))\ell_0) \\
& 
+\sqrt{-1}((\Im(x_{31}^{1}-x_{32}^{2})
-i\Re(x_{31}^{2}-x_{32}^{1}))
+(-\Im(x_{31}^{2}+x_{32}^{1})
+i\Re(x_{31}^{1}+x_{32}^{2}))
\ell_0)], \\
\varPsi(Z)&=
-\sqrt{-1}\frac{\Im(\beta+\gamma)}{2}
-\Im(\alpha)i-\Re(\alpha)\ell_0
-\frac{\Im(\beta-\gamma)}{2}i\ell_0,
\end{aligned}$$
where $X=(x_{31},x_{32})\in\mathfrak g_{-1}$ and 
$Z=(\alpha, \beta,\gamma)
\in\mathfrak g_{-2}$. Here we use the notation 
$x_{3i}=x_{3i}^{(2)}=\begin{bmatrix}
x_{3i}^1 \\ x_{3i}^2 
\end{bmatrix}$, 
where 
$x_{3i}^1$ and $x_{3i}^2$ are $m\times 1$ matrices. 
$\varPsi$ is isomorphic to $\mathfrak g_-$ onto 
$\mathfrak H^{(3)}(\mathbb H,K_{q/2})$ as a pseudo $H$-type Lie algebra. 

From these results, \cite[Theorem 3.6]{AS14:0} and \cite[\S3]{gom96:0}, 
a finite dimensional real SGLA $\gla s$ 
whose negative part is isomorphic to a pseudo $\Div H$-type Lie algebra 
of the third class 
is the prolongation of $\mathfrak s_-$ and is one of 
the following : 

\medskip
\begin{tabular}{|c|c|c|c|}\hline
$\mathbb F$ & $\sgn \innerproduct_{-2}$ & $\mathfrak s$ & the gradation \\ \hline\hline
$\mathbb H$ & $(3,1)$ & $\mathfrak{su}(q+2,2)$ & $({\rm (AIIIa)}_{l,2},\{\alpha_2,\alpha_{l-1}\})$ \\ \hline
$\mathbb H'$ & $(1,3)$ & $\mathfrak{su}(p,p)$ & $({\rm (AIIIb)}_{l},\{\alpha_2,\alpha_{l-1}\})$ \\ \hline
$\mathbb O$ & $(7,1)$ & ${\rm EIII}$ & $({\rm EIII},\{\alpha_1,\alpha_6\})$ \\ \hline
$\mathbb O'$ & $(3,5)$ & ${\rm EII}$ & $({\rm EII},\{\alpha_1,\alpha_6\})$ \\ \hline
\end{tabular}

\section{Pseudo $H$-type Lie algebras satisfying the  $J^2$-condition}
In this section we first see that a pseudo $H$-type Lie algebra 
is isomorphic to a pseudo $H$-type Lie algebra of the first class 
sketchily. For the details of the proof, 
we refer to \cite{MKM18:0}. 
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra. 
For any $x\in \mathfrak n_{-1}$ with $\inph{x}{x}\ne0$ we set 
$$J_{\mathfrak n_{-2}}(x)=
\{\,J_{z}(x):z\in\mathfrak n_{-2}\,\},\qquad 
\mathfrak n_{-1}(x)=\mathbb Rx+J_{\mathfrak n_{-2}}(x);
$$
then $\mathfrak n_{-1}(x)$ is a non-degenerate subspace of $\mathfrak n_{-1}$ 
with respect to $\innerproduct$. 
We say that  $(\mathfrak n,\innerproduct)$ satisfies the $J^2$ condition if 
for any $z\in \mathfrak n_{-2}$ and any $x\in \mathfrak n_{-1}$ with $\inph{x}{x}\ne0$, $\mathfrak n_{-1}(x)$ is $J_z$-stable. 
Clearly if $\dim \mathfrak n_{-2}=1$, then 
$(\mathfrak n,\innerproduct)$ satisfies the $J^2$-condition. 
If a pseudo $H$-type Lie algebra $(\mathfrak n,\innerproduct)$ 
is equivalent to a pseudo $H$-type Lie algebra $(\mathfrak n',\innerproduct')$  satisfying the $J^2$ condition, 
then $(\mathfrak n,\innerproduct)$ also satisfies one. 
   
Let $(\mathfrak n,\innerproduct)$  be a pseudo $H$-type Lie algebra satisfying the $J^2$-condition. For $x\in \mathfrak n_{-1}$ with $\inph{x}{x}\ne0$ 
we set $\mathcal A_x=\mathbb R\times \mathfrak n_{-2}$; then $\mathcal A_x$ 
is a real vector space. We define a multiplicative operation $\underset{x}{*}$ on $\mathcal A_x$ as follows:  for $(\lambda_1,z_1),(\lambda_2,z_2)\in \mathcal A_x$, we put 
$$(\lambda_1,z_1)\underset{x}{*}(\lambda_2,z_2)=(\lambda_3,z_3),$$
where $(\lambda_3,z_3)$ is defined by 
$$(\lambda_11_{\mathfrak n_{-1}}+J_{z_1})(\lambda_2 1_{\mathfrak n_{-1}}+J_{z_2})x=(\lambda_3 1_{\mathfrak n_{-1}}+J_{z_3})x.$$ 
Then $(\mathcal A_x,+,\underset{x}{*})$ is an algebra over $\mathbb R$. 
We define an endomorphism $s$ of $\mathcal A_x$ as follows:  
$$s(\lambda,z)=(\lambda,-z);$$
then $s$ is an anti-involution of $\mathcal A_x$ and satisfies 
$$(\lambda,z)+s(\lambda,z)=(2\lambda,0)\in\mathbb R, 
\quad (\lambda,z)\underset{x}{*}s(\lambda,z)=(\lambda^2+\inph{z}{z},0)\in\mathbb R.$$ 
We define $N:\mathcal A_x\to\mathbb R$ as follows: 
$$N(\lambda,z)= (\lambda,z)\underset{x}{*}s(\lambda,z);$$ 
then $N$ is a non-degenerate quadratic form on $\mathcal A_x$ and hence 
$(\mathcal A_x,s)$ becomes a Cayley algebra. 

Furthermore we can prove that $\mathcal A_x$ becomes an alternative algebra and hence a normed algebra. By Hurwitz theorem (\cite[Theorem 6.37]{har90:0}), $\mathcal A_x$ is isomorphic to one of 
$\mathbb R,\mathbb C,\mathbb C',\mathbb H,\mathbb H',\mathbb O,\mathbb O'$ 
as a Cayley algebra. 
However since $\mathfrak n_{-2}\ne0$, $\mathcal A_x$ is not isomorphic to $\mathbb R$. 
Also the Cayley algebra $\mathcal A_x$ does not depend on the choice 
of the element $x$. 

We choose elements $x_1,\dots,x_{r+s}$ of $\mathfrak n_{-1}$ satisfying the following conditions: 
$$\begin{aligned}
&\inph{x_i}{x_i}=1\quad  (i=1,\dots,r),\qquad \inph{x_j}{x_j}=-1\quad (j=r+1,\dots,r+s), \\
&
\inph{\mathfrak n_{-1}(x_i)}{\mathfrak n_{-1}(x_j)}=0\quad  (i\ne j), \qquad 
\mathfrak n_{-1}=\mathfrak n_{-1}(x_1)\oplus\cdots\oplus\mathfrak n_{-1}(x_{r+s}).
\end{aligned}$$ 
In particular, if $\mathcal A_{x_i}$ is isomorphic to $\mathbb O$ or $\mathbb O'$ for some $i$, then $r+s=1$. 
We denote by $\mathbb F$ the Cayley algebra $\mathcal A_{x_1}$. 
We define a linear mapping $\varphi$ of $\mathfrak n$ to 
$\mathfrak h^{(1)}(\mathbb F,1_{r,s})=\mathbb F^{r+s}\oplus \ima \mathbb F$ as follows: 
$$\varphi\left(\sum_{i=1}^{r+s}(\lambda_ix_i+J_{z_i}(x_i))\right)
=((\lambda_1,z_1),\dots,(\lambda_{r+s},z_{r+s})) 
\ (\lambda_i\in\mathbb R,z_i\in\mathfrak n_{-2}),\quad 
\varphi(z)=-z \ (z\in\mathfrak n_{-2}). 
$$
Then $\varphi$ is an isomorphism as a pseudo $H$-type Lie algebra. 
\begin{theorem}
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra. 
The following three conditions are mutually equivalent: 
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi}) }
\item $(\mathfrak n,\innerproduct)$ satisfies the $J^2$-condition; 
\item $(\mathfrak n,\innerproduct)$ is of the first class;
\item The cps-FGLA associated with  $(\mathfrak n,\innerproduct)$
is of semisimple type. 
\end{enumerate}
\label{th41}
\end{theorem}
\begin{proof}
The implication (i) $\Rightarrow$ (ii) is obtained from the above result. 
The implication (ii) $\Rightarrow$ (iii) follows from \S3.6. 
Finally we prove the implication (iii) $\Rightarrow$ (i). 
Now we assume the condition (iii). 
From the classification of the prolongations of 
cps-FGLAs of semisimple type, the prolongation of 
$(\mathfrak n,[\innerproduct_{-1}])$ is isomorphic to 
the prolongation of the cps-FGLA associated 
with some pseudo $H$-type Lie algebra of the first class. 
Thus (iii) $\Rightarrow$ (i) follows from Proposition \ref{prop22}.  
\end{proof}
\section{The prolongations of the FGLAs and the cps-FGLAs associated with pseudo $H$ type Lie algebras}
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra, and 
let $\cgla{g}$ be the prolongation of $\mathfrak n$. 
The natural inclusion $\iota$ of 
$\mathfrak{so}(\mathfrak n_{-2},\innerproduct_{-2})$ into 
$\check{\mathfrak g}_0$ is defined by 
$$[\iota(v\wedge u),x]=\frac14[J_v,J_u](x)\ 
(x\in\mathfrak n_{-1}),\quad 
[\iota(v\wedge u),z]=(v\wedge u)(z)\ 
(z\in\mathfrak n_{-2}),$$
where $v\wedge u$ is the skew-symmetric endomorphism 
$\inph{v}{\cdot}u-\inph{u}{\cdot}v$. 

Here we quote useful results from \cite{AS14:0} and \cite{AS14:1}. 
\begin{proposition}[{\cite[Theorem 2.3]{AS14:0}}]
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra, 
and let $\cgla{g}$ be the prolongation of $\mathfrak n$. Then 
$$\check{\mathfrak g}_0=\mathfrak{so}(\mathfrak n_{-2},\innerproduct_{-2})\oplus\mathbb RE\oplus \check{\mathfrak h}_0,$$
where $E$ is the characteristic element of the GLA $\cgla g$ and 
$\check{\mathfrak h}_0=\{~x\in\check{\mathfrak g}_0:
[x,\mathfrak n_{-2}]=0~\}$.  
\label{prop51}
\end{proposition}
Let $(\mathfrak n,\innerproduct)$ and $\cgla g$ be as in Proposition \ref{prop51}.  Moreover let $\gla g$ be the prolongation of $(\mathfrak n,[\innerproduct_{-1}])$. We define subspaces $\mathfrak h_0$, $\check{\mathfrak h}_0^a$ and 
$\check{\mathfrak h}_0^s$ of $\check{\mathfrak g}_0$ as follows:  
$$\begin{aligned}
& \mathfrak h_0=\check{\mathfrak h}_0\cap\mathfrak g_0, \\
& \check{\mathfrak h}_0^a=
\{~D\in\check{\mathfrak h}_0:\inph{[D,x]}{y}+\inph{x}{[D,y]}=0
\quad \text{for all}\ x,y\in \mathfrak n_{-1}~\}, \\
& \check{\mathfrak h}_0^s=
\{~D\in\check{\mathfrak h}_0:\inph{[D,x]}{y}-\inph{x}{[D,y]}=0
\quad \text{for all}\ x,y\in \mathfrak n_{-1}~\}, \\
\end{aligned}$$
\begin{corollary}  
Under the above assumptions, 
$$\mathfrak h_0=\check{\mathfrak h}_0^a, \qquad 
\mathfrak g_0=\mathfrak{so}(\mathfrak n_{-2},\innerproduct_{-2})\oplus\mathbb RE\oplus \check{\mathfrak h}_0^a$$ 
\label{cor1}
\end{corollary}
\begin{proof}
Since $D^\top\in\check{\mathfrak h}_0$ for 
$D\in\check{\mathfrak h}_0$, we get 
$\check{\mathfrak h}_0=\check{\mathfrak h}_0^a\oplus\check{\mathfrak h}_0^s$, so $\mathfrak h_0=\check{\mathfrak h}_0^a$. From Proposition \ref{prop51} 
the last assertion is obvious. 
\end{proof}
\begin{theorem}[{\cite[Theorem 3.1 and Remark 3.2]{AS14:1}}]
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra with 
$\dim\mathfrak n_{-2}\geqq3$, and let $\cgla{g}$ be the prolongation of $\mathfrak n$. 
If $\check{\mathfrak g}_1\ne0$, 
then $\cgla{g}$ is a finite dimensional SGLA. 
\label{th51}
\end{theorem}

Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra with 
$\dim\mathfrak n_{-2}\geqq3$. 
Since a pseudo $H$-type Lie algebra is a real extended translation algebra, 
if the prolongation of $\mathfrak n$ is simple, 
then $\dim \mathfrak n_{-2}=3,4,7$ or $8$ 
(\cite[Theorem 3.6]{AS14:0}). 
Hence by Theorem \ref{th51} we obtain the following

\begin{corollary}  
Let $(\mathfrak n,\innerproduct)$ and $\cgla g$ be as in Theorem \ref{th51}. 
If $\dim \mathfrak n_{-2}\ne 3,4,7,8$, then 
$\check{\mathfrak g}_p=0$ for all $p\geqq1$. 
\end{corollary}
\begin{lemma}
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra, 
and let $\gla{g}$ be the prolongation of $(\mathfrak n,[\innerproduct_{-1}])$. 
For $p\geqq1$, the condition ``$x\in\mathfrak g_p$ and 
$[x,\mathfrak g_{-2}]=0$'' implies $x=0$. 
\label{lem51}
\end{lemma}
\begin{proof}
We identify $\mathfrak h_0$ with a subspace of $\mathfrak{gl}(\mathfrak n_{-1})$.  
For a subspace $\mathfrak a$ of $\mathfrak{gl}(\mathfrak n_{-1})$ 
we denote by $\rho^{(k)}(\mathfrak a)$ the $k$-th (algebraic) prolongation of 
$\mathfrak a$. 
By Corollary \ref{cor1}, 
$\mathfrak h_0\subset \mathfrak{so}(\mathfrak n_{-1},\innerproduct_{-1})$; 
hence 
$\rho^{(1)}(\mathfrak h_0)\subset\rho^{(1)}(\mathfrak{so}(\mathfrak n_{-1},\innerproduct_{-1}))=0$. 
The lemma is proved. 
\end{proof}
\begin{theorem}
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra, 
and let $\gla g$ be the prolongation of $(\mathfrak n,[\innerproduct_{-1}])$. 
If $\mathfrak g_2\ne0$ and if the $\mathfrak g_0$-module $\mathfrak g_{-2}$ 
is irreducible, then $\gla g$ is a finite dimensional SGLA. 
\label{th52}
\end{theorem}
\begin{proof} 
Since the prolongation of a cps-FGLA of semisimple type is simple, 
it suffices to prove that $\mathfrak g$ is semisimple. 
Let $\mathfrak r$ be the radical of $\mathfrak g$. Then $\mathfrak r$ is a graded ideal of $\mathfrak g$. That is, putting $\mathfrak r_p=\mathfrak r\capprod \mathfrak g_p$, we see that $\gla r$. 
Let $\mathfrak t$ be the nilpotent radical $[\mathfrak g,\mathfrak r]$ of 
$\mathfrak g$. Assume that $\mathfrak t\ne0$. 
Since $\mathfrak t$ is a nilpotent ideal of $\mathfrak g$, 
there exists $k$ such that $\mathfrak t^{(k)}:=\mathcal C^k(\mathfrak t)\ne0$ 
and $\mathfrak t^{(k+1)}:=\mathcal C^{k+1}(\mathfrak t)=0$, where 
$(\mathcal C^i(\mathfrak t))_{i\geqq0}$ is the ascending central series of $\mathfrak t$. 
Clearly $\mathfrak t$ and $\mathfrak t^{(k)}$ are graded ideals of $\mathfrak g$; putting $\mathfrak t_p=\mathfrak t\cap\mathfrak g_p$ and 
$\mathfrak t^{(k)}_p=\mathfrak t^{(k)}\cap\mathfrak g_p$, 
we get $\mathfrak t=\bigoplus\limits_{p\in\mathbb Z}\mathfrak t_p$ and 
$\mathfrak t^{(k)}=\bigoplus\limits_{p\in\mathbb Z}\mathfrak t^{(k)}_p$. 
Since $\mathfrak t^{(k)}_{-2}$ is a $\mathfrak g_0$-submodule of 
$\mathfrak g_{-2}$, $\mathfrak t^{(k)}_{-2}=0$ or 
$\mathfrak t^{(k)}_{-2}=\mathfrak g_{-2}$. If 
$\mathfrak t^{(k)}_{-2}=0$, then 
$[\mathfrak t^{(k)}_{-1},\mathfrak g_{-1}]\subset \mathfrak t^{(k)}_{-2}=0$, 
so by non-degeneracy, $\mathfrak t^{(k)}_{-1}=0$. Moreover since 
$[\mathfrak t^{(k)}_{0},\mathfrak g_{-1}]\subset \mathfrak t^{(k)}_{-1}=0$, 
by transitivity, $\mathfrak t^{(k)}_{0}=0$. 
Similarly we see that $\mathfrak t^{(k)}_{p}=0$ for all $p\geqq0$, 
which is a contradiction. 
Next if $\mathfrak t^{(k)}_{-2}=\mathfrak g_{-2}$, then 
$[\mathfrak t_p,\mathfrak g_{-2}]=
[\mathfrak t_p,\mathfrak t^{(k)}_{-2}]\subset \mathfrak t^{(k+1)}=0$. 
By Lemma \ref{lem51} $\mathfrak t_p=0$ for all $p\geqq2$. 
Since $\mathfrak t=[\mathfrak g,\mathfrak r]
\supset [E,\mathfrak r]\supset \bigoplus\limits_{p\ne0}\mathfrak r_p$, 
we obtain $\mathfrak r_p=0$ for all $p\geqq2$. 
Hence $\mathfrak g/\mathfrak r=\bigoplus\limits_{p\in\mathbb Z}
\mathfrak g_p/\mathfrak r_p$ 
is a semisimple GLA such that 
$\mathfrak g_{-2}/\mathfrak r_{-2}=0$ and $\mathfrak g_2/\mathfrak r_2=\mathfrak g_2\ne0$. 
By semisimplicity, we get that 
$\dim\mathfrak g_{-2}/\mathfrak r_{-2}=\dim\mathfrak g_2/\mathfrak r_2$, 
which is a contradiction. 
Thus we obtain that $\mathfrak t=0$. 
As above $\mathfrak r_p=0$ for $p\ne0$ and hence $\mathfrak r=0$. 
Therefore $\mathfrak g$ is semisimple. 
\end{proof}

\begin{theorem}
Let $(\mathfrak n,\innerproduct)$ be a pseudo $H$-type Lie algebra, 
and let $\gla g$ be the prolongation of the associated cps-FGLA 
$(\mathfrak n,[\innerproduct_{-1}])$. 
\begin{enumerate}
\item If $\dim \mathfrak n_{-2}=1$, then 
$\gla g$ is one of finite dimensional SGLAs of types 
$({\rm(AI)}_l,\{\alpha_1,\alpha_l\})$, 
$({\rm(AIIIa)}_{l,p},\{\alpha_1,\alpha_l\})$,  
$({\rm(AIIIb)}_l,\{\alpha_1,\alpha_l\})$,  
$({\rm(AIV)}_l,\{\alpha_1,\alpha_l\})$. 
\item If $\dim \mathfrak n_{-2}=2$, then 
$\gla g$ is not semisimple and $\mathfrak g_2=0$. 

\item Assume that $\dim\mathfrak n_{-2}\geqq3$. 
If $\mathfrak g_{2}\ne0$, then 
$\gla g$ is a finite dimensional SGLA and coincides with the prolongation of $\mathfrak n$.  
Furthermore for 
$\mathfrak g_{2}$ to be nonzero, it is necessary and sufficient 
that $(\mathfrak n,\innerproduct)$ is a pseudo $\Div H$-type Lie algebra of the first class. 
\end{enumerate}
\label{th53}
\end{theorem}
\begin{proof}
(1) Since $\dim \mathfrak n_{-2}=1$, the pseudo $H$-type Lie algebra $\mathfrak n$ 
satisfies the $J^2$-condition. Hence (1) follows from Theorem \ref{th41} and the results of 3.6. 

(2) If $\mathfrak g$ is semisimple, then $\dim\mathfrak g_{-2}\ne2$ 
(Theorem \ref{th41}). Hence $\mathfrak g$ is not semisimple. 
If the $\mathfrak g_0$-module $\mathfrak g_{-2}$ is irreducible 
(resp. reducible), 
then, by Theorem \ref{th52} (resp. by the results of \S3.5), 
we obtain $\mathfrak g_2=0$. 

(3) Assume that $\dim\mathfrak n_{-2}\geqq3$ and $\mathfrak g_{2}\ne0$. 
Then $\check{\mathfrak g}_1\ne0$. 
By Theorem \ref{th51}, $\check{\mathfrak g}$ is a finite dimensional SGLA. 
Let $B$ be the Killing form of $\check{\mathfrak g}$. Then 
$B([\check{\mathfrak h}_0,\check{\mathfrak g}_2],\mathfrak g_{-2})=
B(\check{\mathfrak g}_2,[\check{\mathfrak h}_0,\check{\mathfrak g}_{-2}])=0$. 
By non-degeneracy of the Killing form of $\check{\mathfrak g}$, 
we get $[\check{\mathfrak h}_0,\check{\mathfrak g}_2]=0$. 
Since 
$\mathfrak{so}(\mathfrak n_{-2},\innerproduct_{-2})\subset \mathfrak g_0$, 
by Proposition \ref{prop51} the subspace $\mathfrak g_2$ of 
$\check{\mathfrak g}_2$ is $\check{\mathfrak g}_0$-stable. 
Since the $\check{\mathfrak g}_{0}$-module $\mathfrak g_{-2}$ is irreducible, so is $\check{\mathfrak g}_{2}$. 
Since $\mathfrak g_2\ne0$,  we obtain $\mathfrak g_{2}=\check{\mathfrak g}_{2}$. 
By \cite[Lemma 1.6]{tan79:1}, we see that 
$\mathfrak g_1\supset[{\mathfrak g}_{-1},{\mathfrak g}_2]=[\check{\mathfrak g}_{-1},\check{\mathfrak g}_2]=\check{\mathfrak g}_1$ 
and hence $\check{\mathfrak g}_1=\mathfrak g_1$. 
Also by \cite[Lemma 1.3]{tan79:1} we see that
$\mathfrak g_0\supset[{\mathfrak g}_{-1},{\mathfrak g}_1]=[\check{\mathfrak g}_{-1},\check{\mathfrak g}_1]=\check{\mathfrak g}_0$ 
and hence $\check{\mathfrak g}_0=\mathfrak g_0$. 
By the definitions of the prolongations, we obtain that 
$\check{\mathfrak g}_p=\mathfrak g_p$ for all $p\geqq0$. 
The last assertion follows from Theorem \ref{th41}. 
\end{proof}
\begin{corollary}
Let $(\mathfrak n,\innerproduct^{(1)})$ and 
$(\mathfrak n,\innerproduct^{(2)})$ be two pseudo $H$-type Lie algebras 
whose associated  FGLAs coincide. 
Let $\mathfrak g^{(1)}=\bigoplus\limits_{p\in\mathbb Z}\mathfrak g^{(1)}_p$ 
and $\mathfrak g^{(2)}=\bigoplus\limits_{p\in\mathbb Z}\mathfrak g^{(2)}_p$ 
be the prolongations of $(\mathfrak n,[\innerproduct^{(1)}_{-1}])$ and 
$(\mathfrak n,[\innerproduct^{(2)}_{-1}])$ respectively. 
If $\dim \mathfrak n_{-2}\geqq3$, 
$\mathfrak g^{(1)}_2\ne0$ and $\mathfrak g^{(2)}_2\ne0$,  
then $(\mathfrak n,\innerproduct^{(1)})$ is equivalent to 
$(\mathfrak n,\innerproduct^{(2)})$.  
\end{corollary}
\begin{proof}
By Theorem \ref{th53} (3), 
we obtain that the prolongation $\cgla g$ of $\mathfrak n$ is an SGLA and that 
$\check{\mathfrak g}=\mathfrak g^{(1)}=\mathfrak g^{(2)}$. 
By Proposition \ref{prop22} we see that 
$(\mathfrak n,\innerproduct^{(1)})$ is equivalent to 
$(\mathfrak n,\innerproduct^{(2)})$. 
\end{proof}


