Solution
= Solution
Each <structure constant of a Lie algebra> satisfies $f_{ab}{}^d=-f_{ba}{}^d$, so $f_{abc}=f_{ab}{}^d\kappa_{dc}$ is antisymmetric in $a,b$. Invariance of the <Killing form> gives
$$
f_{abc}=\kappa([T_a,T_b],T_c)
=\kappa(T_a,[T_b,T_c])=f_{bca}.
$$
This cyclic symmetry together with antisymmetry in the first pair implies antisymmetry under every transposition. Thus $f_{abc}$ is totally antisymmetric.