= Adjoint gauge variation for a positive-sign covariant derivative
{title2=$\delta A_\mu=\partial_\mu\omega+ig[A_\mu,\omega]$}
Choose the <gauge covariant derivative> $D_\mu=\partial_\mu+igA_\mu$ and matter transformation $\phi'=h\phi$. Covariance $D'_\mu\phi'=hD_\mu\phi$ forces $A'_\mu=hA_\mu h^{-1}+(i/g)(\partial_\mu h)h^{-1}$. For $h=1-ig\omega+O(\omega^2)$, expansion gives the displayed <adjoint covariant derivative> variation. With an orthonormal Hermitian Lie-algebra basis, $\delta A^a_\mu=\partial_\mu\omega^a-gf^{abc}A^b_\mu\omega^c$. Thus Lorenz-gauge variation is $\delta(\partial\cdot A)=\partial\cdot\mathscr D\omega$. Choosing the <Faddeev-Popov operator> as $\mathcal M=-\partial\cdot\mathscr D$, with its field-independent sign absorbed in normalization, gives ghost action $-\bar c\partial\cdot\mathscr D c$; consistent propagator and vertex signs follow from this choice.
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