Uniqueness of an invariant bilinear form on a simple Lie algebra
= Uniqueness of an invariant bilinear form on a simple Lie algebra
Every invariant bilinear form on a finite-dimensional complex simple Lie algebra is a scalar multiple of its Killing form. A nondegenerate invariant form identifies the algebra with its dual; comparing this identification with the Killing form gives an endomorphism of the irreducible adjoint representation, so <Schur lemma> makes it scalar.