Proposition: No Lie-bracket defect

The derivations ${\delta }_X$ satisfy

on every ordered Wick word, all joint compact kernels, and ${\mathcal{D}}_E$.

Proof. For the scalar smearing, integration by parts gives

Indeed, the transpose of $d_X$ with respect to the flat measure is $-X^{\mu }{\partial }_{\mu }-3{\sigma }_X$. The divergence cocycle gives

Hence the claimed identity holds on every smeared scalar.

It passes to derivatives because the action is prolonged, not assigned a new primary weight. It passes to normal products by Lemma lem:Wcov, which ensures compatibility with contraction relations. It passes to arbitrary ordered words because both sides are derivations. Continuity and support-preserving tensor density then give the complete joint-test identity.