# General gauge algebra identity

In https://arxiv.org/abs/1011.1145 the following rather general gauge algebra identity (2.4) is used

$$\delta_{gct}(\xi) B_\mu^{\>\>A} + \xi^\lambda R_{\mu\lambda}^{\quad\! A} -\sum_{\{C\}}\delta(\xi^\lambda B_\lambda^{\>\>C})B_\mu^{\>\>A} = 0$$

The free $$A$$ index references any chosen gauge transformation in the gauge algebra, $$B_\mu^{\>\>A}$$ its corresponding gauge field / connection, $$R_{\mu\nu}^{\quad\! A}$$ its corresponding field strength / curvature, and $$\xi = \xi^\lambda \partial_\lambda$$ the generator of the general coordinate transformations $$\delta_{gct}(\xi)$$. The last sum over $$C$$ is the sum over all gauge transformations $$\delta_C$$ in the gauge algebra.

What follows is clear (and is not relevant to my post) but the author provides no explanation of this identity nor cites a reference where to look it up. Presumably it is valid for any gauge algebra that closes off-shell? Anyway, this relation is not clear to me. Can anyone please provide a derivation of this identity or a reference where it is derived? Thank you.

$$\begin{eqnarray}\delta_{gct}(\xi) B_\mu^{\>\>A} &\;=\;& \mathcal L_\xi B_\mu^{\>\>A} \\ &=& \xi^\lambda\partial_\lambda B_\mu^{\>\>A} + B_\lambda^{\>\>A}\partial_\mu \xi^\lambda \\ &=& \xi^\lambda\partial_{[\lambda} B_{\mu]}^{\>\>A} + \partial_\mu(\xi^\lambda B_\lambda^{\>\>A})\\ &=& \xi^\lambda \big( \partial_{[\lambda} B_{\mu]}^{\>\>A} + f_{BC}^{\quad A} B_\lambda^{\>\>B}B_\mu^{\>\>C} \big) + \partial_\mu(\xi^\lambda B_\lambda^{\>\>A}) + f_{BC}^{\quad A} B_\mu^{\>\>B}(\xi^\lambda B_\lambda^{\>\>C})\\ &=& \xi^\lambda R_{\lambda\mu}^{\quad \!A} + \delta \big(\epsilon = {\textstyle\sum}_C \xi^\lambda B_\lambda^{\>\>C}) B_\mu^{\>\>A} \end{eqnarray}$$ Notation: Here $$\mathcal L$$ is the Lie derivative, index antisymetrization is normalized as $$T_{[ab]} \equiv T_{ab}-T_{ba}$$, and summation on a contracted internal gauge index $$A, B, C,...$$ means over the entire gauge algebra. So for gauged Poincare the sum on $$B^C$$ runs over the vielbein $$e^a$$ and Lorentz connection $$\omega_a^{\>\,b}$$. The last term happens to be the action of gauge algebra transformation $$\delta(\epsilon)$$ acting on the fixed connection $$B_\mu^{\>\>A}$$, but with the usual gauge parameter $$\epsilon$$ replaced by $$\sum_C\xi^\lambda B_\lambda^{\>\>C}$$.