I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a subspace, homologous to the algebra of the Lorentz group, within the affine space of spin connections. The dimension of the subspace would be then 6. However, I cannot prove this idea myself. Does anybody can help (or correct) me with this? Sorry if the wording of the question is poor, but I am not a specialist in differential geometry.