\ltimes (\mathfrak h_1 \times K^3).$$ In particular, fix the ad-invariant metric $\la \,,\,\ra$ in $({\mathfrak n }_{2,3}, B_{2,3})$ given by the matricial representation as in (\[m3223\]).

The set of skew-symmetric derivations is represented by the Lie algebra $$\Dera({\mathfrak n }_{2,3}) \simeq \mathfrak{sl}(2,K) \ltimes \mathfrak h_1$$ while the set of inner derivations is (isomorphic to) $\hh_1$.

In the next paragraphs we describe $\Der({\mathfrak n }_{2,3})$ and $\Dera({\mathfrak n }_{2,3})$ explicitly as subalgebras of $\mathfrak{gl}(5,K)$ putting emphasis on the actions and representations.

By canonical computations one verifies that an element in $\Der({\mathfrak n }_{2,3})$ has the following matricial representation in the basis $e_1,e_2,e_3,e_4,e_5$

$$\left( \begin{matrix}
 z_1+z_4 &-z_2 & 0 & 0 & 0 \\
-z_3 & z_1 -z_4& 0 & 0 & 0 \\
 z_6& z_5 &2z_1 & 0 & 0 \\
 z_7+z_9 & z_8 & z_5 & 3z_1+z_4 & -z_2\\
 z_{10}& z_7-z_9 & -z_6 & -z_3 & 3z_1-z_4
 \end{matrix}
 \right).$$

As usual, denote by $E_{ij}$ the 5$\times$5 matrix which has a 1 at the place $ij$ and $0$ in the other places.

Consider the vector subspace of dimension three spanned by the matrices $$X = E_{32}+E_{43} \qquad Y= E_{31}- E_{53} \qquad Z=E_{41}+ E_{52}$$ which obey the Lie bracket relation $[X,Y]=Z$, thus 