Let $M$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and $K$ a $(1,1)$ tensor field on $M$. I am trying to prove that if $K$ is parallel then $\nabla_X K(Y) = K(\nabla_XY)$ for every $X,Y$ vector fields on $M$.
I tried to use the definition of covariant derivative for tensor fields but it deals with the covariant derivative of $1$-forms, I mean, the formula is:
$$(\nabla_XK)(Y,\omega) = X(K(Y,\omega)) - K(\nabla_XY,\omega) - K(Y, \nabla_X \omega).$$
How can I conclude the claim?