= Gauge covariance of a scalar covariant derivative
{title2=$\delta_X(D_\mu\phi)=\epsilon R(X)D_\mu\phi$}
With $D_\mu=\partial_\mu+R(A_\mu)$ and $\delta_X A_\mu=-\epsilon\partial_\mu X+\epsilon[X,A_\mu]$, the derivative terms in $\delta_X(D_\mu\phi)$ 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.
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