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We consider Lie algebras admitting an ad-invariant metric, and we study theproblem of uniqueness of the ad-invariant metric up to automorphisms. This is acommon feature in low dimensions, as one can observe in the knownclassification of nilpotent Lie algebras of dimension $\leq 7$ admitting anad-invariant metric. We prove that uniqueness of the metric on a complex Liealgebra $\mathfrak{g}$ is equivalent to uniqueness of ad-invariant metrics onthe cotangent Lie algebra $T^*\mathfrak{g}$; a slightly more complicatedequivalence holds over the reals. This motivates us to study the broader classof Lie algebras such that the ad-invariant metric on $T^*\mathfrak{g}$ isunique.  We prove that uniqueness of the metric forces the Lie algebra to be solvable,but the converse does not hold, as we show by constructing solvable Liealgebras with a one-parameter family of inequivalent ad-invariant metrics. Weprove sufficient conditions for uniqueness expressed in terms of both theNikolayevsky derivation and a metric counterpart introduced in this paper.  Moreover, we prove that uniqueness always holds for irreducible Lie algebraswhich are either solvable of dimension $\leq 6$ or real nilpotent of dimension$\leq 10$.

Uniqueness of ad-invariant metrics