Lemma

For past-compact smooth $u$, ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}u=u$. For compactly supported $w$ and past-compact $S$, $\int w\cdot {{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}S=\int {{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot S$. For a covariant $v$ and a density $T$, one of them compactly supported, $\int v\cdot {\mathcal{L}}_{\xi }T=-\int {\mathcal{L}}_{\xi }v\cdot T$.

Proof. ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}u=\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}\square \mathcal{P}u=\mathcal{PP}u=u$, since $\mathcal{P}$ is constant and ${{\mathsf{E}}_0}^{\mathrm{R}}\square v=v$ for past-compact $v$. The second statement is the transposition of ${{\mathsf{E}}_0}^{\mathrm{R}}$, both integrals being over the compact set $\mathrm{supp}{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cap \mathrm{supp}S$. The third: $v\cdot T$ is a scalar density, ${\mathcal{L}}_{\xi }(v\cdot T)={\partial }_{\lambda }({\xi }^{\lambda }v\cdot T)$ integrates to zero, and ${\mathcal{L}}_{\xi }$ is a derivation.