the family of free nilpotent and free metabelian nilpotent Lie algebras.

[**Theorem**]{} \[t1\]. [*Let ${\mathfrak n }_{m,k}$ be the free $k$-step nilpotent Lie algebra in $m$ generators. Then ${\mathfrak n }_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$.*]{}

The techniques for the proof do not make use of the extension procedures mentioned before, but properties of free nilpotent Lie algebras which combined with the ad-invariance condition enable the deduction of the Lie algebras ${\mathfrak n }_{2,3}$ and ${\mathfrak n }_{3,2}$. We note that for $k=2,3$ the free and free metabelian $k$-step nilpotent Lie algebras coincide. For the free metabelian case the first approach lies in the fact that 2-step solvable Lie algebras admitting ad-invariant metrics are nilpotent and at most 3-step (Lemma \[le11\]). Thus working out we get the next result.

[**Theorem**]{} \[t2\]. [*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 $(m,k)=(2,3)$.*]{}

These two Lie algebras have been studied since a long time in sub-Riemannian geometry [@Mo]. Thus ${\mathfrak n }_{2,3}$ is associated to the Carnot group distribution (see for instance [@BM]), which is related 