I am using these supergravity lecture notes by Gary W. Gibbons. On page 18, the author claims that geodesics and autoparallels coincide for a theory with totally antisymmetric torsion, and proves it by using the following identity (which he states without proof):
$$\Gamma_{\mu\;\;\nu}^{\;\;\rho}=\{_{\mu\;\;\nu}^{\;\;\rho}\}+K_{\mu\;\;\nu}^{\;\;\rho},$$
where
$$K_{\mu\rho\nu}=-\frac{1}{2}(T_{\mu\rho\nu}+T_{\rho\mu\nu}+T_{\rho\nu\mu}).$$
Here, $\Gamma_{\mu\;\;\nu}^{\;\;\rho}$ are the coefficients of an arbitrary connection with totally antisymmetric torsion and $\{_{\mu\;\;\nu}^{\;\;\rho}\}$ are the coefficients for the Levi-Civita connection, while $T_{\rho\nu\mu}$ is the torsion. Is there a proof of this identity somewhere?