Remark: Infinitesimal form

Setting $\zeta =0$ and taking the $\lambda$-linear part of the family $\exp (\lambda \kappa {\mathcal{L}}_{\xi })$, so that ${\delta }^{(0)}h=K\xi$ and ${\delta }^{(1)}h={\mathcal{L}}_{\xi }h$ in the notation $q=\eta +\kappa h$ with $\kappa$-dependent $h$, (eq:X1law)–(eq:X2law) and (eq:X2affine) read

with ${{\delta }^{(1)}}_{\xi }{\mathsf{X}}_1={\mathsf{X}}_1[{\mathcal{L}}_{\xi }h]$ and ${{\delta }^{(0)}}_{\xi }{\mathsf{X}}_2={{\mathsf{X}}_2^{\prime }}_h[K\xi ]$. These are the two coefficient identities named in the plan. The second is the main calculation; it is proved by Theorem thm:coordinate and verified directly in Proposition prop:audit.