Diffeomorphism identity for a metric action (${\nabla }_a{E}^{ab}\equiv 0$)

For a diffeomorphism-invariant pure metric action, write $\delta S=-\int \sqrt{-g}{E}^{ab}\delta g_{ab}$. Substitute $\delta g_{ab}={\nabla }_a{\xi }_b+{\nabla }_b{\xi }_a$ for an arbitrary compactly supported vector field and integrate by parts. Symmetry of $E^{ab}$ gives ${\nabla }_a{E}^{ab}=0$ without using field equations. This Noether identity concerns the gravitational variational tensor alone.