# Invariance of gauge-fixing condition in background field method

In the Peskin & Schröder (chapter 16.6) they use the background field method and spilt the gauge field into an background field $$A$$ and a fluctuation field $$\mathcal{A}$$. Next they claim that the Lagrangian (16.98) is invariant under the background gauge symmetry $$A^a_\mu\rightarrow A^a_\mu+D_\mu\beta^a\tag{16.99a}$$ $$\mathcal{A}^a_\mu\rightarrow \mathcal{A}^a_\mu-f^{abc}\beta^b\mathcal{A}_\mu^a\tag{16.99b}$$

(+ transformations of ghosts and fermions) where $$D$$ is the covariant derivative constructed from the background field. However I don’t see why the gauge fixing term $$D^\mu\mathcal{A_\mu^a}$$ should be invariant under this transformation.

1. One trick is to rewrite the (infinitesimal) background gauge transformation (16.99) as \begin{align} \delta_B A_{\mu} ~=~& [D_{\mu},\beta],\cr \delta_B D_{\mu} ~=~& -i[D_{\mu},\beta],\cr \delta_B {\cal A}_{\mu} ~=~& -i[{\cal A}_{\mu},\beta], \end{align} \tag{16.99} using the conventions (15.44) & (16.91). Here $$[\cdot,\cdot]$$ denotes a commutator or a Lie bracket depending on context.
2. The Lorenz gauge function $$[D^{\mu},{\cal A}_{\mu}]$$ therefore transforms covariantly $$\delta_B [D^{\mu},{\cal A}_{\mu}] ~\stackrel{(16.99) + \text{Jac.id.}}{=}~ -i[[D^{\mu},{\cal A}_{\mu}],\beta].$$
3. The gauge-fixing term $$\kappa\left([D^{\mu},{\cal A}_{\mu}],~[D^{\nu},{\cal A}_{\nu}]\right)$$ is invariant under the background gauge transformation (16.99) because the Killing form $$\kappa(\cdot,\cdot)$$ is associative/invariant.