If $(M,J)$ is an almost complex manifold, $\mathcal{N}$ (Nijenhuis tensor) is: $$\mathcal{N}(X,Y)=[JX,JY]-J[X,JY]-J[JX,Y]-[X,Y].$$
I am trying to compute it in local coordinates $(x_{1},\cdots,x_{n})$.
I have tried to compute each Lie bracket separately.
If $X=\sum_{i=1}^{n}X_{i}\frac{\partial}{\partial x_{i}}$ and $Y=\sum_{i}^{n}Y_{i}\frac{\partial}{\partial x_{i}}$, then $$[X,Y]=\sum_{i,j=1}^{n}(X_{i}\frac{\partial}{\partial x_{i}}(Y_{j})-Y_{i}\frac{\partial}{\partial x_{i}}(X_{j}))\frac{\partial}{\partial x_{j}}.$$
Now, I am trying to compute $[JX,Y]$, but I have troubles.
By applying the formula of the Lie bracket, $$[JX,Y]=\sum_{i,j=1}^{n}((JX)_{i}\frac{\partial}{\partial x_{i}}(Y_{j})-Y_{i}\frac{\partial}{\partial x_{i}}((JX)_{j}))\frac{\partial}{\partial x_{j}}.$$
For $f\in C^{\infty}(M)$, $$JX(f)=J(X(f))=\sum_{i=1}^{n}J_{i}\frac{\partial}{\partial x_{i}}(\sum_{j=1}^{n}X_{j}\frac{\partial}{\partial x_{j}}(f))=\sum_{i=1}^{n}J_{i}\sum_{j=1}^{n}\frac{\partial}{\partial x_{i}}(X_{j}\frac{\partial}{\partial x_{j}}(f))=$$$$\sum_{i=1}^{n}J_{i}\sum_{j=1}^{n}(\frac{\partial}{\partial x_{i}}(X_{j})\frac{\partial f}{\partial x_{j}}+X_{j}\frac{\partial}{\partial x_{i}}(\frac{\partial}{\partial x_{j}}(f))).$$
Now, I don't know which is $(JX)_{i}$ since I have vector fields evaluated in different smooth functions... Can anyone help me, please? Or is there another way of computing $[JX,Y]$?
In local coordinates, $J = J_i^kdx^i\otimes\partial_k$ so that
$$J(\partial_l) = J_i^k dx^i(\partial_l) \otimes \partial_k = J_i^k\delta_l^i\otimes \partial_k = J_l^k\otimes \partial_k = J_l^k\partial_k.$$
Therefore, if $X = X^l\partial_l$, then
$$JX = J(X^l\partial_l) = X^lJ(\partial_l) = X^lJ_l^k\partial_k.$$
So $JX = (JX)^k\partial_k$ where $(JX)^k = J_l^kX^l$.