Proposition: Classical action identity on the frozen algebra

For every $\xi \in {\mathfrak{g}}_U$,

as a formal variational identity.

Proof. Every current in (eq:switchedNoether) is compactly supported in the gauge-support region. Thus the spatial and temporal boundary integrals vanish.

On a neighborhood of this region $g=1$ and $dg=0$. Therefore

All retained terms in (eq:switchedNoether) vanish. The discarded term from varying ${\kappa }^{2}g{\ell }_2$ with $\kappa {{\delta }_{\xi }}^{(1)}$ is of order ${\kappa }^3$.