the algebraic relations simply carry over, so we have for the Ward operators generating background gauge transformations: $$\begin{aligned}
\label{b-ward-d-d-new}[\widetilde{\widetilde{{\cal W}^{\rm B}_{\rm D}}}(v_1), \widetilde{\widetilde{{\cal W}^{\rm B}_{\rm D}}}(v_2)] &=& \widetilde{\widetilde{{\cal W}^{\rm B}_{\rm D}}}([v_1, v_2]) - {\cal W}^{\rm B}_{\rm YM}(v_1 v_2\cdot \bar{F})\\
\label{b-ward-l-l}[{\cal W}^{\rm B}_{\rm YM}(\lambda_1), {\cal W}^{\rm B}_{\rm YM}(\lambda_2)] &=& {\cal W}^{\rm B}_{\rm YM}(f \lambda_1\lambda_2)\\
\label{b-ward-d-l}[\widetilde{\widetilde{{\cal W}^{\rm B}_{\rm D}}}(v), {\cal W}^{\rm B}_{\rm YM}(\lambda)] &=& 0\end{aligned}$$

As for the true gauge transformations, the merely [*background*]{} field dependent parameter $\lambda^a= \bar{A}^a_\mu v^\mu$ of the compensating ${\sf SU(N)}_{\rm loc}$ transformation is not subject to true gauge transformations. The algebra that generates these transformations can be easily computed therefore by taking advantage of the linearity of commutators. We obtain $$\begin{aligned}
\label{g-ward-d-d-new}[\widetilde{\widetilde{{\cal W}^{\rm G}_{\rm D}}}(v_1), \widetilde{\widetilde{{\cal W}^{\rm G}_{\rm D}}}(v_2)] &=& \widetilde{\widetilde{{\cal W}^{\rm G}_{\rm D}}}([v_1, v_2]) + {\cal W}^{\rm G}_{\rm YM}(v_1 v_2\cdot \bar{F})\\
\label{g-ward-l-l}[{\cal W}^{\rm G}_{\rm YM}(\lambda_1), {\cal W}^{\rm G}_{\rm YM}(\lambda_2)] &=& {\cal W}^{\rm G}_{\rm YM}(f \lambda_1\lambda_2)\\
\label{g-ward-d-l}[\widetilde{\widetilde{{\cal W}^{\rm G}_{\rm D}}}(v), {\cal W}^{\rm G}_{\rm YM}(\lambda)] &=& {\cal W}^{\rm G}_{\rm YM}(v\cdot\bar{\nabla} \lambda)\end{aligned}$$

By writing $\widetilde{\widetilde{W}}$ we distinguish these Ward operators from the modified one that was defined with respect to the undecomposed fields; in addition, their notation shall remind us of the fact that the gauge field entered their 