The Nijenhuis tensor is defined to be:
$$(1):\quad N_J(X,Y)\equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY], $$
for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:
$$(2):\quad J:TM\rightarrow TM\quad|\quad J^2=-I_{TM}. $$
The requirement is to show that given $J$ is integrable, $N_J$ is vanishing. I am having an issue with the following computation which gives a vanishing $N_J$, regardless of whether $J$ is integrable or not:
$$ N_J(X,Y)\equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY] $$ $$ =[X,Y]+J^2XY-JYJX+JXJY-J^2YX-[JX,JY] $$ $$ =[X,Y]+J^2[X,Y] $$ $$ =[X,Y]-[X,Y]=0\quad\text{(by (2) alone)}. $$ I find this very strange because this makes absolutely no reference at all to $J$ being integrable or not, rather this follows solely from the definition of $J$ in $(2)$. Suggesting that if $J$ is an almost complex structure, then $N_J$ is always vanishing. Is it that I can write $(2)$ only for integrable $J$ ? If someone could please explain what is it that I am not doing right or do not understand correctly ?...