inv} = \{F\:|\:\widetilde{{\cal W}_{\rm D}}(v) F = 0 \:\wedge\: {\cal W}_{\rm YM}(\lambda) F = 0 ~ \forall ~ v^\mu,\:\lambda^a\} \end{aligned}$$

The algebra relations receive extra contributions now since the Ward operators act on the field dependent parameters of the transformations as well. This leads to an algebra of the desired form (with $(v_1 v_2 \cdot F)^a\equiv v_1^\mu v_2^\nu F_{\mu\nu}^a$): $$\begin{aligned}
\label{ward-d-d-new}[\widetilde{{\cal W}_{\rm D}}(v_1), \widetilde{{\cal W}_{\rm D}}(v_2)] &=& \widetilde{{\cal W}_{\rm D}}([v_1, v_2]) - {\cal W}_{\rm YM}(v_1 v_2\cdot F)\\
\label{ward-l-l-new}[{\cal W}_{\rm YM}(\lambda_1), {\cal W}_{\rm YM}(\lambda_2)] &=& {\cal W}_{\rm YM}(f \lambda_1\lambda_2)\\
\label{ward-d-l-new}[\widetilde{{\cal W}_{\rm D}}(v), {\cal W}_{\rm YM}(\lambda)] &=& 0\end{aligned}$$

If we now split the dynamical fields into fluctuations and background configurations, we have to decide whether the field dependent transformation parameter in (\[new-diff\]) should contain the full or the background gauge field only. Since, as already mentioned, the metric and gauge field fluctuations do not enter the final form of $S_{\rm gh}^{(2)}$ anyhow, we may safely opt for the latter already at this point:

$$\widetilde{\widetilde{{\cal W}_{\rm D}}}\!\!\!\raisebox{0.15cm}{{\scriptsize B,G}}(v)={\cal W}^{\rm B,G}_{\rm D}(v)+{\cal W}^{\rm B,G}_{\textrm{YM}}(\bar A \cdot v)$$

According to the usual distinction between true and background gauge transformations, we now have to consider two classes of Ward operators as well. In the background case, 