Covariant gauge-fixing density as a BRST variation (${\mathcal{L}}_{\mathrm{gf}+\mathrm{gh}}=s[-{\bar{c}}^a({\partial }^{\mu }{A}_{\mu }^a+\xi b^a/2)]$)

For the left graded Leibniz rule and $s\bar{c}=-b$, $sb=0$, the displayed gauge-fixing fermion gives $b\partial \cdot A+\xi b^2/2+\bar{c}\partial \cdot Dc$. The ghost signs follow from odd parity. BRST nilpotence implies invariance of this density without using field equations. Eliminating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$ for nonzero $\xi$.