First Bianchi identity
for any three vector fields .
Using repeated simplication and the Jacobi identity
Let us plug the definition of the Riemann curvature tensor:
This can be regrouped as:
Now, since is torsion-free, we have and similar simplifications yield:
again using the fact that is torsion-free, this simplifies to:
this becomes zero by the Jacobi identity.
Using the differential Bianchi identity
Fill this in later