ad-invariance property also says that $$0=\la e_4, e_3\ra =\la e_4, e_4\ra= \la e_4, e_5\ra=\la e_5, e_3\ra=\la e_5, e_5\ra.$$ Moreover $$\la e_1, e_3\ra= \la e_1, [e_1, e_2]\ra =0 \quad \mbox { and } \quad \la e_2, e_3\ra=\la e_2, [e_1, e_2] \ra=0;$$ $$\la e_1, e_4\ra=\la e_1, [e_1, e_3]\ra=0 \quad \mbox{ and } \quad \la e_2, e_5\ra=\la e_2, [e_2, e_3]\ra=0.$$ Therefore $$\la e_3, e_3\ra=\la [e_1,e_2], e_3\ra=-\la e_2, e_4\ra= \la e_1, e_5\ra =\alpha \neq 0,$$ which amounts to the following matrix for $\la \,,\,\ra$ in the ordered basis above: $$\label{m23}
\left(
\begin{matrix}
a_{11} & a_{12} & 0 & 0 & \alpha \\
a_{12} & a_{22} & 0 & - \alpha & 0 \\
0 & 0 & \alpha & 0 & 0 \\
0 & -\alpha & 0 & 0 & 0 \\
\alpha & 0 & 0 & 0 & 0 
\end{matrix}
\right)
\qquad \mbox{ with } a_{ij} \in K, \forall i,j=1,2 \mbox{ and } \alpha \neq 0.$$

Remark \[23\], Corollary \[corofree\] and the previous theorem imply the next result.

\[t2\] Let $\tilde{{\mathfrak n }}_{m,k}$ be the free metabelian $k$-step nilpotent Lie algebra in $m$ generators. Then $\tilde{{\mathfrak n }}_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$.

The free 