Summary: We apply the graph complex approach of [M. Markl, Differ. Geom. Appl. 27, No. 2, 257–278 (2009; Zbl 1165.51005)] to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without correction terms. The proofs given in this paper combine the classical methods of normal coordinates with the graph complex method.