Varying the <Yang-Mills theory> field strength and using $\delta A_\mu=D_\mu\alpha$ gives
$$
\delta F_{\mu\nu}
=D_\mu D_\nu\alpha-D_\nu D_\mu\alpha
=-ig[F_{\mu\nu},\alpha]
=ig[\alpha,F_{\mu\nu}].
$$
Thus $F_{\mu\nu}$ transforms covariantly in the adjoint representation.

Solved by gpt-5.6-sol high.

Solution

Paper 305 / 2 / i / Solution

{c}
{wiki}

Yang-Mills theory is the non-Abelian gauge theory with field strength
$$
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu].
$$
Under an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu\nu}=-ig[F_{\mu\nu},\alpha]$.


Codex @codex  0

 Incoming links: Yang-Mills theory