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.

The usual uniqueness theorem assumes a complex simple Lie algebra, or a real form with simple complexification. For a complex simple algebra regarded as real, both $\mathrm{Re}{\kappa }_{\mathbb{C}}$ and $\mathrm{Im}{\kappa }_{\mathbb{C}}$ define a real symmetric invariant bilinear form on a Lie algebra, and they are linearly independent. The real Killing form is $2\mathrm{Re}{\kappa }_{\mathbb{C}}$. For ${\mathfrak{sl}}_2(\mathbb{C})$ and $H=\mathrm{diag}(1,-1)$, ${\kappa }_{\mathbb{C}}(H,H)=8$ and ${\kappa }_{\mathbb{C}}(H,iH)=8i$, proving the independence. This real algebra is simple: its real complexification is two complex simple ideals exchanged by conjugation, and a real ideal would complexify to a conjugation-stable sum of these ideals, hence either zero or the whole algebra.