Proposition: Transformation of the retained mixed term; not an invariance statement

Put

For $\xi \in {\mathfrak{g}}_U$, its exact finite change in ${\mathfrak{A}}_0$ is

Equivalently, it is

The first two terms are linear in the shift parameter and the last is quadratic.

Proof. Both $c_1[h]$ and ${\mathsf{L}}_{1,g}[h]$ are linear in $h$, while $c_1[K\xi ]=D_{\xi }f$. Expanding $-{\mathsf{L}}_{1,g}[h+K\xi ]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}(c_1[h]+D_{\xi }f)$ produces the first term in (eq:mixed-transformation) and two terms containing $-{\mathsf{Q}}_{K\xi }{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. For a compact test $u$,

The last summand is zero in ${\mathfrak{A}}_0$. Apply this identity with $u=c_1[h]$ and $u=D_{\xi }f$. The integral form follows from ${D_{\xi }}^{\mathsf{T}}=-X_{\xi }$.