Definition: Formal adjoint and self-adjoint subspace

Write $E^{\mu \nu }[h]={\sum }_{a\le b,\beta }{K^{\mu \nu }}_{ab,\beta }{\partial }^{\beta }{h}_{ab}$, where $K$ depends on the external jets according to the reading. Under Gate V the formal adjoint is

with derivatives hitting the external coefficients retained (Leibniz on ${\sigma }_{\epsilon }$- and ${\chi }_{\mathrm{I}}$-jets, unless these are constant in the reading). Then $\int ⟨t,E[h]⟩=\int ⟨E^{†}[t],h⟩$ for compactly supported $t$. The self-adjoint subspace is $\mathrm{SA}:=\{C\in {\mathcal{C}}_{\mathrm{coh}}:E_C={E_C}^{†}\}$, as an identity in formal jets. By Lemma lem:joint, $\mathrm{SA}={⨁}_D{\mathrm{SA}}_D$.