Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 4 / d / Solution

The Killing form is invariant under the adjoint action:Together with $F_{\mu \nu }^{\prime }={\mathrm{Ad}}_g{F}_{\mu \nu }$, this immediately givesso every positive integral power ${\mathcal{L}}_1$ is gauge invariant.

The adjoint covariant derivative obeysThis follows either by substituting the transformation laws or by applying the covariance of $D_{\mu }^{\prime }$ to an adjoint-valued field. A second use of invariance of the Killing form giveswhich proves gauge invariance of ${\mathcal{L}}_2$.