I am trying to understand the proof that the Levi-Civita connection is torsion free. In the notes in theorem 6.8 it is written that
$g(\nabla_XY, Z) - g(\nabla_YX, Z) = g(Z, [X,Y])$
proves that connection is torsion free. My question is how do we show that the above relation satisfies the 0 torsion definition
$\nabla_XY - \nabla_YX = [X, Y]$?