Killing-form invariance under a Lie-group adjoint action ($\kappa ({\mathrm{Ad}}_{g}X,{\mathrm{Ad}}_{g}Y)=\kappa (X,Y)$)

Every Lie algebra automorphism $S$ obeys ${\mathrm{ad}}_{SX}=S{\mathrm{ad}}_X{S}^{-1}$. The cyclic property of the trace therefore makes the Killing form invariant under $S$. For a matrix Lie group, $S={\mathrm{Ad}}_g$ 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.