Integrability of almost complex structure If we want to check an integrability of an almost complex structure in $R^{4}$ is it enough to take vectors $X=X^{1}\frac{\partial}{\partial x^{1}}$ and $Y=Y^{1}\frac{\partial}{\partial x^{1}}$ and then calculate Nijenhuis tensor $N(X,Y)$, or we must calculate $N$ on vectors $X=X^{1}\frac{\partial}{\partial x^{1}}+X^{2}\frac{\partial}{\partial x^{2}}+X^{3}\frac{\partial}{\partial y^{1}}+X^{4}\frac{\partial}{\partial y^{2}}$ and $Y=Y^{1}\frac{\partial}{\partial x^{1}}+Y^{2}\frac{\partial}{\partial x^{2}}+Y^{3}\frac{\partial}{\partial y^{1}}+Y^{4}\frac{\partial}{\partial y^{2}}$?
Thank you!
 A: As pointed out in the comments, the Nijenhuis tensor is a tensor so $N(fX, Y) = fN(X, Y)$ and $N(X, fY) = fN(X, Y)$ for any smooth function $f$. So we can rewrite the $N(X, Y)$ as
\begin{align*}
&\color{white}{+} X^1Y^1N(\partial_{x^1}, \partial_{x^1}) + X^1Y^2N(\partial_{x^1}, \partial_{x^2}) + X^1Y^3N(\partial_{x^1}, \partial_{y^1}) + X^1Y^4N(\partial_{x^1}, \partial_{y^2})\\
& + X^2Y^1N(\partial_{x^2}, \partial_{x^1}) + X^2Y^2N(\partial_{x^2}, \partial_{x^2}) + X^2Y^3N(\partial_{x^2}, \partial_{y^1}) + X^2Y^4N(\partial_{x^2}, \partial_{y^2})\\
&+ X^3Y^1N(\partial_{y^1}, \partial_{x^1}) + X^3Y^2N(\partial_{y^1}, \partial_{x^2}) + X^3Y^3N(\partial_{y^1}, \partial_{y^1}) + X^3Y^4N(\partial_{y^1}, \partial_{y^2})\\
&+ X^4Y^1N(\partial_{y^2}, \partial_{x^1}) + X^4Y^2N(\partial_{y^2}, \partial_{x^2}) + X^4Y^3N(\partial_{y^2}, \partial_{y^1}) + X^4Y^4N(\partial_{y^2}, \partial_{y^2}).\\
\end{align*} 
So if the Nijenhuis tensor vanishes on all pairs of coordinate vector fields, then it vanishes on all pairs of vector fields.
