the isometry group of a naturally reductive pseudo-Riemannian space (in particular symmetric spaces) can be endowed with an ad-invariant metric.

Semisimple Lie algebras are examples of Lie algebras admitting an ad-invariant metric since the Killing form is nondegenerate. In the solvable case, the Killing form is degenerate so one must search for another bilinear form with the ad-invariance property. The first investigations concerning general Lie algebras with ad-invariant metrics appeared in [@FS; @MR]. They get structure results proposing a method to construct these Lie algebras recursively. This enables a classification of nilpotent Lie algebras admitting ad-invariant metrics of dimension $\leq 7$ in [@FS] and a determination of the Lorentzian Lie algebras in [@Me]. The point is that by this recursive method one can reach the same Lie algebra starting from two non-isomorphic Lie algebras. This fact difficulties the classification in higher dimensions. More recently a new proposal for the classification problem is presented in [@KO] and this is applied in [@Ka] to get the nilpotent Lie algebras with ad-invariant metrics of dimension $\leq 10$.

However the basic question whether a non-semisimple Lie algebra admits such a metric is still opened. In the present paper we deal with this problem in 