When can the Killing form be put in the form $\kappa_{ij}=-\kappa\delta_{ij}$, and thus be treated as an Euclidean dot product? Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}$ is a Cartan subalgebra. We know when the lie algebra is semi simple then the killing form, $\kappa(X,Y) = Tr(X^{ad}Y^{ad})$, is non degenerate in $\mathfrak{g}$ and more importantly in $\mathfrak{h}$. So it is a bilinear form analogous to the metric providing an isomorphism between $\mathfrak{h}$ and $\mathfrak{h}*$ and allowing us to take the inner product across these spaces.
One thing that isn't clear fromt the literature I've read (and from my Universities lecture notes) is when we are able to to put the killing form in the form $\kappa_{ij} = -\kappa \delta_{ij}$ and so we can treat the inner product  as the normal euclidean dot product. Some notes I have read say this is only true for compact groups, others I have read say you can always do it and some don't even mention it (my lecture notes).
 A: Call a semisimple Lie algebra $\mathfrak{g}$ compact if its Killing form is negative definite. Then the simply connected Lie group $G$ with Lie algebra $\mathfrak{g}$ has its adjoint representation on $\mathfrak{g}$ preserving the Killing form; its image lies in the Lie subgroup of $\text{Aut}(\mathfrak{g})$ preserving the Killing form, which is an orthogonal group $O(\mathfrak{g})$ and in particular which is compact. It follows that the adjoint form $G/Z(G)$ of $G$ (the image of the Killing form) is compact, and hence if the center of $G$ is finite that $G$ is compact. (I think this is always true with these hypotheses but I'm not sure how to prove it.)
The semisimple Lie algebras which arise in this way are exactly the compact real forms of the semisimple complex Lie algebras, e.g. $\mathfrak{su}(n), \mathfrak{so}(n), \mathfrak{sp}(n)$ and the exceptionals.
In the noncompact case the Killing form may have indefinite signature; for example, the Killing form on $\mathfrak{sl}_2(\mathbb{R})$ has signature $(2, 1)$, and the adjoint representation induces an exceptional isomorphism $\mathfrak{sl}_2(\mathbb{R}) \cong \mathfrak{so}(2, 1)$. This is the split analogue of the exceptional isomorphism $\mathfrak{su}(2) \cong \mathfrak{so}(3)$, and both of these isomorphisms complexify to the exceptional isomorphism $\mathfrak{sl}_2(\mathbb{C}) \cong \mathfrak{so}_3(\mathbb{C})$.
A: It is true for compact Lie algebra (Lie algebras of compact simply connected groups) for which Killing is negative definite.
https://en.wikipedia.org/wiki/Killing_form#Connection_with_real_forms
