Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 4 / a / Solution

Substituting $\delta A_{\mu }=g^{-1}{\partial }_{\mu }\alpha -i[A_{\mu },\alpha ]$ into the curvature and collecting terms gives the covariant transformationEquivalently, $\delta F_{\mu \nu }^a=f^{abc}{F}_{\mu \nu }^b{\alpha }^c$ in the stated conventions. Since $F_{\mu \nu }^a{F}^{\mu \nu ,a}$ is proportional to $\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })$, cyclicity of the matrix trace givesThus the Yang-Mills action is gauge-invariant.