Gauge covariance of a scalar covariant derivative (${\delta }_X(D_{\mu }\varphi )=ϵR(X)D_{\mu }\varphi$)

With $D_{\mu }={\partial }_{\mu }+R(A_{\mu })$ and ${\delta }_X{A}_{\mu }=-ϵ{\partial }_{\mu }X+ϵ[X,A_{\mu }]$, the derivative terms in ${\delta }_X(D_{\mu }\varphi )$ cancel. The remaining terms combine using $R([X,A_{\mu }])=[R(X),R(A_{\mu })]$, so the covariant derivative transforms in the same Lie algebra representation as the scalar. This convention absorbs the gauge coupling into the connection.