TITLE: Proof that the Nijenhuis tensor vanishes in a complex manifold
QUESTION [7 upvotes]: I'm in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any two vector fields $X,Y$ one has
$$
J[X,Y]=J[X,JY]+J[JX,Y]+[X,Y]
$$
I tried applying all the definitions of commutator I actually know, but I can't manage it...
===
Edit: Am I using the wrong definition of commutator? Can you please confirm me that $[X,Y]$ is defined for $X=X_i\partial_i$, $Y=Y_j\partial_j$ to be the vector field
$$
\sum_{i=1}^N\Big(\sum_{j=1}^N X_j\frac{\partial Y_i}{\partial x_j}-Y_j\frac{\partial X_i}{\partial x_j}\Big)\partial_i
$$

REPLY [6 votes]: "Kobayashi-Nomizu's Foundations states the result in the form of theorem 2.5 page 124, and they define the Nijenhuis tensor in "my" way, upto a factor 2"
It's in the volume II, anyway from this book follow that
[JX, JY]= [X, Y]+ J[X, JY]+ J[JX, Y] (for a complex strucure J)
in your expression above on the left you write the term J[X, Y] instead [JX, JY].