A spin connection $A_{ab\mu}=-A_{ba\mu}$ has 24 components. The number of independent components for a flat spin connection can be counted by subtracting the constrains set by the condition of null curvature \begin{equation} R^a_{\phantom{a}b\mu\nu} = 0. \end{equation} The curvature tensor has at most 36 independent components. However, the second Bianchi identities \begin{equation} D_{\rho}R^a_{\phantom{a}b\mu\nu} + D_{\mu}R^a_{\phantom{a}b\nu\rho} + D_{\nu}R^a_{\phantom{a}b\rho\mu} = 0 \end{equation} with $D_{\mu}$ a Weitzenböck covariant derivative, sets 24 constrains in the components of the curvature, leaving only 12 independent. The first Bianchi identities \begin{equation} D_{\rho}T^a_{\phantom{a}\mu\nu} + D_{\mu}T^a_{\phantom{a}\nu\rho} + D_{\nu}T^a_{\phantom{a}\rho\mu} = R^a_{\phantom{a}\rho\mu\nu} + R^a_{\phantom{a}\mu\nu\rho} + R^a_{\phantom{a}\nu\rho\mu} \end{equation} only sets constraints for the Torsion, so there are not additional constraints for the curvature.
The number of independent components of the flat spin connection, after subtracting the constrains, is 12.
I can not be sure if this argument is complete and I cannot found help in the literature. I would appreciate feedback or a useful reference.
Thank for your help!
