A spin connection exists because of the existence of the complex circular basis. One does not have to have a tangent spacetime superimposed on a Riemannian manifold, as was Cartan’s original idea. In three dimensions, the existence of (1), (2), (3) superimposed on X, Y, Z, is enough to produce a spin connection, torsion and curvature, either in Cartan or Riemann. Donald Reed (Advances in Chemical Physics, volume 119(3)) refers to Harry Moses as having discovered the (1), (2), (3) basis, but it is mentioned in Brian Silver’s well known book:

B. L. Silver, “Irreducible Tensorial Methods” (Academic, 1976)

I discovered the (1), (2), (3) basis independently in the nineties, notably in Physica B 1992 and following papers. It is well known that Cartan is internally consistent. The two proofs of the structure equations in note 140(11) should be studied carefully.

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