Let $(M,J)$ be an almost complex manifold and $\nabla$ be a connection on $TM$. I am trying to see how we can obtain an explicit formula for $\nabla_X J$. I know that the way to extend $\nabla$ to a connection on $T^*M$ is by defining for $\alpha \in \Gamma(T^*M)$, $\nabla_X(\alpha)(Y):= X(\alpha(Y)) - \alpha(\nabla_X Y)$, and on $T^*M \otimes TM$ we can extend $\nabla$ to $\nabla_X (\alpha \otimes Y):= (\nabla_X \alpha) \otimes Y + \alpha \otimes (\nabla_X Y)$.
Further, $J \in \Gamma(\textrm{End}(TM))$, so actually $J \in \Gamma(T^*M \otimes TM)$. But I can't see how to deduce from the above extensions an actual formula for $\nabla_X J(\alpha \otimes Y)$ in terms of just $J, \alpha, \nabla_X Y, X, Y$.
One of the problems with this is that I don't know how to write $J$ as an element in $T^*M \otimes TM$ except in local coordinates, where things get ugly. Is there any way to avoid local coordinates? How would a derivation of an expression for $\nabla_X J$ go in any case?
for any vector field $Y$ and covector field $\alpha$, $\alpha(J(Y))$ is a smooth function. Since the directional derivative $\nabla_X$ commutes with contraction, we have : \begin{align} X(\alpha(J(Y)) &= \nabla_X(\alpha(J(Y))) \\ &= (\nabla_X \alpha) (J(Y)) + \alpha((\nabla_XJ)(Y)) + \alpha(J(\nabla_XY)) \end{align} Therefore $\nabla_X J$ is the section of $\operatorname{End}(TM)$ such that : $$\alpha((\nabla_XJ)(Y)) = X(\alpha(J(Y)) - (\nabla_X \alpha) (J(Y)) - \alpha(J(\nabla_XY)) $$