The significance of the tetradic formalism appear in the Einstein–Cartan formulation of general relativity. The tetradic formalism of the theory is more fundamental than its metric formulation as one can not convert between the tetradic and metric formulations of the fermionic actions despite this being possible for bosonic actions . This is effectively because Weyl spinors can be very naturally defined on a Riemannian manifold and their natural setting leads to the spin connection. Those spinors take form in the vielbein coordinate system, and not in the manifold coordinate system.