Remark

On $G_{\mathrm{lin}}[u]=0$, (eq:cov2) is the formula $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }u]=-2{\mathcal{G}}_2(K\xi ,u)$ quoted in X-E's closing remark of E4. Off it the term ${\mathcal{L}}_{\xi }{G}_{\mathrm{lin}}[u]$ is present. For the free gauge-fixed graviton, $P_{\mathrm{gr}}\hat{h}=0$ gives $G_{\mathrm{lin}}[\hat{h}]=\frac{1}{2}\mathcal{P}KC[\hat{h}]$ by (eq:xe-gf), which does not vanish unless $C[\hat{h}]=0$; the shortcut therefore does not apply to $\hat{h}$, and the exact form is used below.