The 1 + 3 covariant equations, embedded in an extended tetrad formalism and describing a spacetime with an arbitrary energy–momentum distribution, are reconsidered. It is shown that, provided the 1 + 3 splitting is performed with respect to a generic time-like congruence with a tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman–Penrose framework
