This is a Theorem made in Lectures on Kahler Geometry by A.Moroiano.
Theorem 5.5 A hermitian metric $h$ on an almost complex manifold $(M,J)$ is kahler iff $J$ is parallel w.r.t levi-civita connection $\nabla$ of $h$.
I am having trouble to understand one of the statement. He is trying to show $Kahler\implies \nabla J=0$. He used a fact involving $\{J,\nabla_XJ\}=J\nabla_X J+(\nabla_X J)J=0$ for any $X\in TM$. I think I am supposed to show for any $X,Y,Z\in TM, h(\{J,\nabla_XJ\}Y,Z)=0$ but I do not know how to achieve this.
Why is $J,\nabla_XJ$ anti-commuting? It seems this part is obvious and I could not see this.
Hint Differentiate both sides of the defining identity $J \circ J = -\operatorname{id}_{TM}$ with $\nabla_X$.