Solution
= Solution
Since $(\operatorname{ad}_{T_a})^c{}_b=f_{ab}{}^c$, the <Killing form> has components
$$
\boxed{\kappa_{ab}=f_{ac}{}^d f_{bd}{}^c}.
$$
This is the matrix trace of $\operatorname{ad}_{T_a}\operatorname{ad}_{T_b}$.
= Solution
Since $(\operatorname{ad}_{T_a})^c{}_b=f_{ab}{}^c$, the <Killing form> has components
$$
\boxed{\kappa_{ab}=f_{ac}{}^d f_{bd}{}^c}.
$$
This is the matrix trace of $\operatorname{ad}_{T_a}\operatorname{ad}_{T_b}$.