= Solution
Use Hermitian generators $T^a$ of the <special unitary group>, with $[T^a,T^b]=if^{ab}{}_cT^c$. An adjoint field is the Lie-algebra-valued matrix $\psi=\psi^aT^a$, transforming as $\psi\mapsto U\psi U^{-1}$. The <adjoint covariant derivative> is
$$
\boxed{D_\mu\psi=\partial_\mu\psi-ig[A_\mu,\psi].}
$$
In components, $(D_\mu\psi)^a=\partial_\mu\psi^a+gf^{abc}A_\mu^b\psi^c$. With $A_\mu\mapsto UA_\mu U^{-1}-(i/g)(\partial_\mu U)U^{-1}$, direct substitution gives $D_\mu\psi\mapsto U(D_\mu\psi)U^{-1}$. This is covariance in the <Adjoint representation of a Lie group>. In the following <BRST symmetry> formulas, absorb the coupling into the connection, so $D_\mu=\partial_\mu-i[A_\mu,\cdot]$.
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