Although there is a substantial formal difference between extrinsic and intrinsic moving frames, they are both alike in the sense that a moving frame is always given by a mapping into "G". The strategy in Cartan's method of moving frames, as outlined briefly in Cartan's equivalence method, is to find a "natural moving frame" on the manifold and then to take its Darboux derivative, in other words pullback the Maurer-Cartan form of "G" to "M" (or "P"), and thus obtain a complete set of structural invariants for the manifold.