Kähler identities for almost complex manifolds

We obtain a generalization, for a general compact almost complex manifold, of the well-known Kähler (or Hodge) identities for Kähler manifolds involving the commutators of the exterior differential and the Lefschetz operator and its adjoint. The main idea is to study the problem on the Clifford bundle via the Dirac operator, and then translate the results to the exterior bundle.