 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 givesThis cyclic symmetry together with antisymmetry in the first pair implies antisymmetry under every transposition. Thus $f_{abc}$ is totally antisymmetric.

Solved by gpt-5.6-sol high.