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The very definition of an Einstein metric implies that all its geometry isencoded in the Weyl tensor. With this in mind, in this paper we derivehigher-order Bochner type formulas for the Weyl tensor on a four dimensionalEinstein manifold. In particular, we prove a second Bochner type formula which,formally, extends to the covariant derivative level the classical one for theWeyl tensor obtained by Derdzinski in 1983. As a consequence, we deduce someintegral identities involving the Weyl tensor and its derivatives on a compactfour dimensional Einstein manifold.

Bochner type formulas for the Weyl tensor on four dimensional Einstein manifolds