F1: exact covariance with the Lie term retained

The exact identity is the F1 mechanism of X-F XF (Exact second-order covariance identity). The shortcut $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }r]=-2G_2[K\xi ,r]$ requires $G_{\mathrm{lin}}[r]=0$. A gauge-fixed free solution need not satisfy that stronger equation.