Remark: Switch derivatives have not disappeared

Both lemmas are identities for the complete divergence-form operators. They do not replace ${\partial }_{\mu }(g{a_1}^{\mu \nu }{\partial }_{\nu })$ by $g{\partial }_{\mu }({a_1}^{\mu \nu }{\partial }_{\nu })$, nor ${\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \beta }{\partial }_{\beta })$ by ${\sigma }_{\epsilon }{\partial }_{\alpha }({a_1}^{\alpha \beta }{\partial }_{\beta })$. The terms with $\partial g$ in (eq:V1g)–(eq:V2g) and with ${\sigma }_{\epsilon }^{\prime }$ in (eq:X1eq)–(eq:X2eq) remain part of the operators; they are covariant because they vanish where the transformation acts.