Remark

If $\xi$ is a Killing field of $\eta$, both sides of (eq:key) vanish: $K\xi =0$, and $P_{\mathrm{gr}}$ has constant coefficients and is Poincaré covariant. For $\xi \in {\mathfrak{g}}_U$ neither side vanishes in general; this is the statement, already in X-E, that $\hat{h}\mapsto \hat{h}+\kappa {\mathcal{L}}_{\xi }\hat{h}$ is not an automorphism of the free gauge-fixed algebra. The present sheet does not transform $\hat{h}$.