= Killing-form invariance under a Lie-group adjoint action
{c}
{title2=$\kappa(\operatorname{Ad}_gX,\operatorname{Ad}_gY)=\kappa(X,Y)$}
Every Lie algebra automorphism $S$ obeys $\operatorname{ad}_{SX}=S\operatorname{ad}_XS^{-1}$. The <cyclic property of the trace> therefore makes the <Killing form> invariant under $S$. For a matrix <Lie group>, $S=\operatorname{Ad}_g$ is conjugation by $g$, proving the displayed identity even when the group is disconnected. This directly proves <gauge invariance> of Killing-form contractions of adjoint fields.
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