Let $(M,I,h)$ be a Kahler manifold. We know that the complex structure $I$ must be parallel in the sense that $\nabla I = 0$, where $\nabla$ is the Levi-Civita connection on $M$.
I have understood this as follows: for every $Z,W \in \mathcal{C}^{\infty}(TM) $, we have $$ (\nabla_{Z} I)W = \nabla_{Z} (IW) = 0. $$ But for every such $ Z $ and $ W $, we may take $ X = -IW $ such that $ IX = -I^{2}W = W $, $ X \in \mathcal{C}^{\infty}(TM) $, such that $$ (\nabla_{Z}I)X = \nabla_{Z}(IX) = \nabla_{Z}W = 0. $$ Therefore $\nabla$ vanishes everywhere on $\mathcal{C}^{\infty}(TM) \times \mathcal{C}^{\infty}(TM) $.
What is the dumb thing that I am not seeing or understanding wrong?
Many thanks in advance.
I think I noticed where my mistake is. In fact, we have $$ \nabla I = [\nabla, I] $$ over $ I \in \text{End}(TM) $.