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We present structural properties of Lie algebras admitting symmetric,invariant and nondegenerate bilinear forms. We show that these properties arenot satisfied by nilradicals of parabolic subalgebras of real split forms ofcomplex simple Lie algebras, neither by 2-step nilpotent Lie algebrasassociated with graphs, with only few exceptions. These rare cases are,essentially, abelian Lie algebras, the free 3-step nilpotent Lie algebra on2-generators and the free 2-step nilpotent Lie algebra on 3-generators.

Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms