Riemannian curvature two-form ($R\in {\Omega }^2(M;\mathrm{End}(TM))$)

For the Levi-Civita connection, the curvature form of a connection is the endomorphism-valued two-formIt has tensoriality in all three arguments. Its first Bianchi identity follows from torsion-freeness: the cyclic sum is ${\sum }_{\mathrm{cyc}}({\nabla }_X[Y,Z]-{\nabla }_{[Y,Z]}X)={\sum }_{\mathrm{cyc}}[X,[Y,Z]]=0$ by the Jacobi identity.