An intrinsic and exterior form of the Bianchi identities

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis. We demonstrate the equivalence of this new formulation to both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds and codimension one foliations are studied as examples of its utility.