Assuming a Levi-Civita connection exists it is uniquely determined. Using $\nabla g = 0$ and the symmetry of the metric tensor $g$ we find:
$ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z - \nabla_Z X, Y) + g(\nabla_Y Z - \nabla_Z Y, X). $
I am not sure how this is concluded. Can anyone elaborate further on here? I just can't see the link between left side and right side...
The Levi-Civita connection satisfies
$$ X \, g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z). $$
This, together with the symmetry of $g$, is sufficient to check that relation.