& \begin{array}{cc}0&0\end{array} \\
\begin{array}{cc}t_{41}&t_{42}\\t_{51}&t_{52}\end{array} &\begin{array}{cc}t_{11}t_{32}-t_{31}t_{12}\\t_{21}t_{32}-t_{31}t_{22}\end{array} & \det(A) A\end{array}
\right),$$ where $A\in \GL(2,K)$.

Consider the following matrices in $\Aut({\mathfrak n }_{2,3})$: $$\mathcal G=\left\{ \tilde{A}=\left(\begin{array}{ccc}
A &                                \begin{array}{cc}0\\0\end{array} & \begin{array}{cc}0&0\\0&0\end{array} \\
\begin{array}{cc} 0&0\end{array} &        \det(A)                                & \begin{array}{cc}0&0\end{array} \\
\begin{array}{cc}0&0\\0&0\end{array} &\begin{array}{cc}0\\0\end{array} &\det(A)A\end{array}
\right), \, A\in \GL(2,K)\right\}$$ $$\mathcal H=\left\{ h_{(x,y,z)}=\left(
\begin{array}{ccccc}
 1&0&0&0&0 \\
0&1&0&0&0\\
y&x&1&0&0\\ 
z+\frac{1}{2} x y&\frac{1}{2} x^2&x&1&0\\
-\frac{1}{2} y^2&z-\frac{1}{2} xy&-y&0&1\end{array}\right),\, (x,y,z)\in K^3\right\}$$

$$\mathcal R= \left\{ r_{(u,v,w)}=\left(
\begin{array}{ccccc}
 1&0&0&0&0 \\
0&1&0&0&0\\
0&0&1&0&0\\ 
v&w&0&1&0\\
u&-v&0&0&1\end{array}\right),\, (u,v,w)\in K^3\right\}.$$

Note that $\mathcal{G}$, $\mathcal{R}$ and $\mathcal{H}$ are subgroups of $\Aut({\mathfrak n }_{2,3})$.

Moreover every $t$ of the form (\[t\]) can be written as a product of matrices $$t=\tilde{A}\cdot r_{(u,v,w)} \cdot h_{(x,y,z)},\qquad \mbox{ with } \tilde{A} \in \mathcal G,\; r_{(u,v,w)}\in \mathcal R,\; h_{(x,y,z)}\in \mathcal H$$

where $x=t_{32}/\det(A)$, $y=t_{31}/\det(A)$, $z=(t_{22}t_{41}-t_{12}t_{51}+t_{52}t_{11}-t_{42}t_{21})/2\det(A)^2$ and

$u=(2t_{51}t_{11}-2t_{41}t_{21}+t_{31}^2)/2 \det A^2$, $v=(t_{22}t_{41}-t_{31}t_{32}-t_{12}t_{51}+t_{42}t_{21}-t_{52}t_{11})/2 \det A$, $w=(2t_{42}t_{22}-t_{32}^2-2t_{52}t_{12})/2\det A^2$.

The elements of $\mathcal{H}$ commute with those of $\mathcal{R}$; also $\mathcal H \cap  \mathcal R=\{I\}$. Hence $\mathcal{R}\cdot \mathcal{H}\simeq \mathcal{R}\times \mathcal{H}.$

It holds $h_{(x,y, z)} \cdot h_{(x',y',z')}=h_{(x+x', y+ y', z + z' + \frac12 (xy' -x' y))}$, where $\cdot$ denotes the product of matrices. Thus the map from the Heisenberg Lie group $H_1$ to $\Aut({\mathfrak n }_{2,3})$ given by $(x,y, z) \mapsto h_{(x,y,z)}$ is an isomorphism of groups.

Analogously $(u,v,w)\mapsto r_{(u,v,w)}$ 