= Solution
The pure gravitational action is invariant under <diffeomorphisms>. An infinitesimal diffeomorphism generated by a compactly supported <vector field> $\xi$ changes the metric by its <tensor Lie derivative>,
$$
\delta g_{ab}=\mathcal L_\xi g_{ab}=\nabla_a\xi_b+\nabla_b\xi_a.
$$
Using symmetry of $E^{ab}$ and the variational expression already derived,
$$
0=\delta_\xi S_g=-2\int\sqrt{-g}\,E^{ab}\nabla_a\xi_b\,d^4x
=2\int\sqrt{-g}\,(\nabla_aE^{ab})\xi_b\,d^4x.
$$
As $\xi_b$ is arbitrary, the <off-shell metric divergence identity> is
$$
\boxed{\nabla^aE_{ab}=0.}
$$
This <Noether identity> follows for every <metric tensor> without imposing either the gravitational field equation or the matter equations. It is a consequence of the metric action's diffeomorphism invariance, not a conclusion requiring a long component calculation.
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