\mbox { for all }i,j=4,5,6.$$

The ad-invariance property says $$\la e_4, e_1\ra=\la [e_1,e_2], e_1\ra=0$$ and similarly $\la e_4,e_2\ra=0$, therefore $\la e_4,e_3\ra \neq 0$. Analogously, $\la e_5, e_2\ra\neq 0$ and $\la e_6, e_1\ra\neq 0$. Moreover $$\alpha=\la [e_1, e_2], e_3\ra=-\la e_2, e_5, \ra=\la e_1, e_6\ra.$$

Thus in the ordered basis $\{e_1,e_2,e_3,e_4,e_5,e_6\}$, a matrix of the form $$\label{m32}
\left(
\begin{matrix}
a_{11} & a_{12} & a_{13} & 0 & 0 & \alpha \\
a_{12} & a_{22} & a_{23} & 0 & -\alpha & 0 \\
a_{13} & a_{23} & a_{33} & \alpha & 0 & 0 \\
 0 & 0  & \alpha & 0 & 0 & 0 \\
 0 & -\alpha &  0 & 0 & 0 & 0 \\
 \alpha & 0 & 0 & 0 & 0 & 0
 \end{matrix}
 \right) \qquad \mbox{ with } a_{ij} \in K, \forall i,j=1,2,3 \mbox{ and } \alpha \neq 0$$

corresponds to an ad-invariant metric on ${\mathfrak n }_{3,2}$.

The Lie algebra ${\mathfrak n }_{2,3}$ has a basis $\{e_1,e_2,e_3,e_4,e_5\}$ with non-zero Lie brackets $$\label{b23}
[e_1,e_2]=e_3,\qquad [e_1,e_3]=e_4,\qquad [e_2,e_3]=e_5.$$ Let $\la \,,\,\ra$ be an ad-invariant metric on ${\mathfrak n }_{2,3}$. Then $\zz({\mathfrak n }_{3,2})=C^2({\mathfrak n }_{2,3})=span\{e_4, e_5\}$ while $C^1({\mathfrak n }_{2,3})=C^2({\mathfrak n }_{2,3}) \oplus K e_3$.

The 