Solution
= 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 $[H,E_\pm]=\pm2E_\pm$ and $[E_+,E_-]=H$, the Killing matrix in the ordered basis $(E_+,E_-,H)$ is
$$
\begin{pmatrix}0&4&0\\4&0&0\\0&0&8\end{pmatrix}.
$$
This Cartan-Weyl basis is adapted to the root decomposition, but it is not Killing-orthogonal because $\kappa(E_+,E_-)=4$.