& 0 \\
1 & 0 & 0 & 0 & 0 
\end{matrix}
\right).$$

The automorphism groups of ${\mathfrak n }_{3,2}$ and ${\mathfrak n }_{2,3}$
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Here we study the automorfisms of the Lie algebras in Theorem \[t1\]. This is indeed a topic of active research (see for instance [@DF; @DG] and references therein). Our goal is to write explicitly the algebraic structure, in terms of the actions and representations of the different subgroups or subalgebras. We also distinguish the subgroup of orthogonal automorphism (resp. the Lie algebra of skew-symmetric derivations) in presence of the ad-invariant metric fixed in (\[m3223\]).

Recall that a derivation of a Lie algebra $\mathfrak g$ is a linear map $t: \mathfrak g \to \mathfrak g$ satisfying $$t[x,y]=[tx, y] + [x, ty ] \qquad \mbox{ for all } x,y \in \mathfrak g.$$

Whenever $\mathfrak g$ is endowed with a metric $\la \,,\,\ra$ a skew-symmetric derivation of ${\mathfrak g }$ is a derivation $t$ such that $$\label{antisim}
\la t a, b\ra=-\la a, tb\ra \quad \text{ for all }\; a,b\in{\mathfrak g }.$$

We denote by $\Der({\mathfrak g })$ the Lie algebra of derivations of ${\mathfrak g }$, which is the Lie algebra of the group of automorphisms of 