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.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / c / Solution

{title2=$f_{ab}{}^c$}
{wiki=Structure_constants}

For a basis $(T_a)$ of a Lie algebra, its structure constants are defined by $[T_a,T_b]=f_{ab}{}^cT_c$.


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 Incoming links: Structure constant of a Lie algebra