 Solution

The complexification of a Lie algebra is $\mathfrak{su}(2{)}_{\mathbb{C}}=\mathfrak{su}(2){\otimes }_{\mathbb{R}}\mathbb{C}\cong {\mathfrak{sl}}_2(\mathbb{C})$. With $[H,E_{\pm }]=\pm 2E_{\pm }$ and $[E_{+},E_{-}]=H$, the Killing matrix in the ordered basis $(E_{+},E_{-},H)$ isThis Cartan-Weyl basis is adapted to the root decomposition, but it is not Killing-orthogonal because $\kappa (E_{+},E_{-})=4$.