Riemann tensor symmetries

Question

The Riemann tensor has its component expression: $R^{\mu}_{\nu\rho\sigma}=\partial_{\rho}\Gamma^{\mu}_{\sigma\nu}-\partial_{\sigma}\Gamma^{\mu}_{\rho\nu}+\Gamma^{\mu}_{\rho\lambda}\Gamma^{\lambda}_{\sigma\nu}-\Gamma^{\mu}_{\sigma\lambda}\Gamma^{\lambda}_{\rho\nu}.$

It is straight forward to prove the antisymmetry of $R$ in the last two indices; but how to prove the antisymmetry in the first two ones without assuming symmetric connection/torsion-free metric?

Answer 1

This simple change in definition,

$$\nabla_a\nabla_b-\nabla_b\nabla_a$$

for

$$\nabla_a\nabla_b-\nabla_b\nabla_a+T^d_{\quad{ab}}\nabla_d$$

However, that the Riemann tensor with torsion is no longer symmetric under exchange of the first pair of indices with the second.