= Uniqueness of an invariant bilinear form on a simple Lie algebra
Every <invariant bilinear form on a Lie algebra> $B$ on a finite-dimensional complex <simple Lie algebra> $\mathfrak g$ is a scalar multiple of its <Killing form> $\kappa$. Since $\kappa$ is <nondegenerate>, there is a unique <endomorphism> $T$ with $B(x,y)=\kappa(Tx,y)$, even when $B$ itself is degenerate. Invariance of both forms gives $T[z,x]=[z,Tx]$. Thus $T$ intertwines the <Adjoint representation of a Lie algebra>, which is an <irreducible representation> because its <invariant subspaces> are <ideals of a Lie algebra>. The <Schur lemma> gives $T=cI$, hence $B=c\kappa$. In particular, every nonzero such form is automatically <nondegenerate> and symmetric.
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