is an isomorphism of groups from $K^3$ to $ \mathcal R\subseteq \Aut({\mathfrak n }_{2,3})$.

The action by conjugation of $\mathcal{G}$ preserves both $\mathcal{H}$ and $\mathcal{R}$. Thus the map $$\tau_{\tilde{A}}(r,h)=(\tilde{A} r\tilde{A}^{-1},\tilde{A} h\tilde{A}^{-1}).$$ defines a group homomorphism from $\mathcal{G}$ to $\Aut(\mathcal R \times \mathcal{H})$.

The subgroup $\mathcal{R}\times \mathcal{H}$ is normal in $\Aut({\mathfrak n }_{2,3})$ and $\mathcal{G}\cap (\mathcal{R}\times \mathcal{H})=\{I\}$, hence $\Aut({\mathfrak n }_{2,3})\simeq \mathcal G\ltimes_\tau (\mathcal{R}\times \mathcal{H})$ ([@Kn]). It is clear that $\mathcal{G}\simeq \GL(2,K)$, $\mathcal{R}\simeq K^3$ and $\mathcal{H}\simeq H_1$ and these isomorphisms preserve the action of $\GL(2,K)$ in $H_1$ and $K^3$. So the next result follows.

\[pro5\] The group of automorphisms of ${\mathfrak n }_{2,3}$ is $$\Aut({\mathfrak n }_{2,3}) \simeq \GL(2,K)\ltimes (K^3 \times H_1),$$ where $H_1$ denotes the Heisenberg Lie group of dimension three.

The aditional orthogonal condition leads the following result.

The group of orthogonal automorphisms of $({\mathfrak n }_{2,3},B_{2,3})$ is $$\Auto({\mathfrak n }_{2,3}) \simeq \mathcal S \ltimes H_1,$$ where $\mathcal S$ is the subgroup of $\mathcal G$ consisting of the matrices $\tilde{A}$ with $A\in GL(2,K),\,\det(A)=\pm1$. The action in the semidirect product is the restriction of the action of $\mathcal G$ in the Heisenberg Lie group $H_1$ described before.

The Lie algebra of derivations of ${\mathfrak n }_{2,3}$ is isomorphic to $$\mathfrak{gl}(2,K) 