this is a faithful representation of the Heisenberg Lie algebra $\hh_1$.

Also denote by $$U= E_{42}\qquad V=E_{41}-E_{52}\qquad W=E_{51}$$ a basis of the vector subspace which spans an abelian Lie algebra of dimension three.

The algebra $\mathfrak h_1 \times K^3$ is an ideal of the Lie algebra of derivations namely the radical. Denote by $E,F,H, T$ the matrices given by $$\begin{array}{rclrcl}
E & = & -E_{12}-E_{45} & F & = & -E_{21}-E_{54} \\
H & = & E_{11}-E_{22} + E_{44}-E_{55} & T & = & E_{11}+E_{22}+2E_{33}+3E_{44}+2E_{55}.
\end{array}$$

Thus $span \{E,F,H\}\simeq \mathfrak{sl}(2,K)$ and therefore $span \{E,F,H,T\}\simeq \mathfrak{gl}(2,K)=\mathfrak{sl}(2,K)\times K$. The action of $\mathfrak{gl}(2,K)$ on $\mathfrak h_1\times K^3$ preserves each of the ideals $\mathfrak h_1$ and $K^3$ respectively.

The action of $\mathfrak{sl}(2,K)$ on $\mathfrak h_1$ is is given by derivations, that is $t \in \mathfrak{sl}(2, K) \simeq \Der(\hh_1)$, in the basis $X,Y,Z$, is represented by $$\left(
\begin{matrix}
h & e & 0\\
f &  -h &  0 \\
0 &  0  & 0
\end{matrix}
\right).$$

The action of $\mathfrak{sl}(2,K)$ on $K^3$ is given by its irreducible representation in dimension three (see [@Hu]) $$\ad(eE+fF+hH)|_{U,V,W} = \left( \begin{matrix}
2h &  2e &  0\\
f & 0 &  e\\
0 & 2f & -2h
\end{matrix}
\right).$$

On 