Curvature homogeneity for four-dimensional manifolds

Let (M,g) be an n-dimensional, connected Riemannian manifold with Levi Civita connection $\nabla$ and Riemannian curvature tensor R defined by $$ R_XY = [\nabla_X, \nabla_Y] - \nabla_{[X,Y]} $$ for all smooth vector fields X, Y. $\nablaR, \cdots, \nabla^kR, \cdots$ denote the successive covariant derivatives and we assume $\nabla^0R = R$.