Proposition: Key identity

For every smooth vector field $\xi$ and smooth symmetric $u$,

In the flat gauge of X-E, ${\mathcal{F}}^{(1)}\equiv 0$ and ${{\mathcal{Q}}^{(1)}}_{\mathrm{flat}}[K\xi ]={\mathcal{G}}_2(K\xi ,\cdot )=\frac{1}{2}[{\mathcal{L}}_{\xi },G_{\mathrm{lin}}]=[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]-[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]$.

Proof. As in Lemma lem:cov2, reduce to compactly supported $\xi$ and let $F_s$ be its flow. Lemma lem:natural gives $\mathcal{F}[F_s^{\ast }u;F_s^{\ast }\eta ]=F_s^{\ast }\mathcal{F}[u;\eta ]$. Differentiate at $s=0$ using (eq:F-expansion) with $\kappa h_{\mathrm{cl}}\to sK\xi$: ${\mathcal{F}}^{(1)}[K\xi ]u+{\mathcal{F}}^{(0)}{\mathcal{L}}_{\xi }u={\mathcal{L}}_{\xi }{\mathcal{F}}^{(0)}u$. The second identity follows from the first, (eq:cov2), (eq:Q1), and (eq:F0); or directly from the third identity of Lemma lem:natural. The flat-gauge statement is (eq:cov2) with (eq:F0).