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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.