Curvature commutator on a contravariant tensor
= Curvature commutator on a contravariant tensor
For a type-$(2,0)$ tensor, the <Ricci identity> acts once on each contravariant index:
$$
[\nabla_\alpha,\nabla_\beta]T^{\mu\nu}
=R^\mu{}_{\rho\alpha\beta}T^{\rho\nu}
+R^\nu{}_{\rho\alpha\beta}T^{\mu\rho}.
$$