Non-Abelian gauge transformation ($A_{\mu }\mapsto R{A}_{\mu }{R}^{-1}-g^{-1}({\partial }_{\mu }R)R^{-1}$)

For the convention $D_{\mu }={\partial }_{\mu }+g{A}_{\mu }$, a gauge group transformation $\Phi \mapsto R\Phi$ acts on the gauge potential by $A_{\mu }\mapsto R{A}_{\mu }{R}^{-1}-g^{-1}({\partial }_{\mu }R)R^{-1}$. The gauge field strength transforms by conjugation. This makes $D_{\mu }\Phi$ transform in the same gauge group representation as $\Phi$.