= Killing-normalized contraction of a cubic trace tensor
{c}
{title2=$c_a{}^{mn}B_{mnb}=-\frac12H_{ab}$}
Lowering and raising <structure constants> with the <Killing form> gives $c_a{}^{mn}c_{mn}{}^r=-\delta_a{}^r$. Since $c_a{}^{mn}$ is antisymmetric in $m,n$, only the <commutator> part of the <cubic trace tensor of a Lie algebra representation> contributes, proving $c_a{}^{mn}B_{mnb}=-H_{ab}/2$. In an adapted compact basis with $\kappa_{ab}=-\delta_{ab}$ and $H_{ab}=-\mu\delta_{ab}$, the resulting sign is positive: $+\mu\delta_{ab}/2$. Defining the lowered structure constants with a positive Euclidean metric instead would reverse this sign.
// Target: lie-theory.bigb
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