 Torsion correction to the Dirac square

For a metric-compatible connection with torsion tensor $T^{\rho }{}_{\mu \nu }={\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \mu }^{\rho }$, expand the square using Clifford multiplication and the full affine second derivative. Its antisymmetric part is spin curvature minus $T^{\rho }{}_{\mu \nu }{\nabla }_{\rho }$. This gives the displayed formula. The curvature need not have the torsion-free pair symmetry or first Bianchi identity, so its Clifford contraction need not reduce to a scalar $-R/4$. General torsion produces additional first-order and nonscalar curvature couplings.