= Real-simple exception to uniqueness of the Killing form
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 $\operatorname{Re}\kappa_{\mathbb C}$ and $\operatorname{Im}\kappa_{\mathbb C}$ define a real symmetric <invariant bilinear form on a Lie algebra>, and they are linearly independent. The real <Killing form> is $2\operatorname{Re}\kappa_{\mathbb C}$. For $\mathfrak{sl}_2(\mathbb C)$ and $H=\operatorname{diag}(1,-1)$, $\kappa_{\mathbb C}(H,H)=8$ 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.
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