I'm reading proposition 4.16 from Ballmann's Lectures on Kähler Manifolds:
Let $M$ be a complex manifold with a compatible Riemann metric $g$ and Levi-Civita connection $\nabla$, then: $$d\omega(X,Y,Z)=g((\nabla_XJ)Y,Z)+g((\nabla_YJ)Z,X)+g((\nabla_ZJ)X,Y)$$ $$2g((\nabla_XJ)Y,Z)=d\omega(X,Y,Z)-d\omega(X,JY,JZ)$$
He begins the proof by saying "Since $M$ is a complex manifold, we can assume that the vector fields $X,Y,Z,JY$ and $JZ$ commute".
I assume "X,Y commutes" (two variables) means $[X,Y]=0$, but I don't understand what it has to do with $M$ being complex and what does he mean by "$X,Y,Z,JY,JZ$ commute" (five variables).
The equalities you have to prove are of the form $T(X,Y,Z)=0$, where $T$ is a tensor. A tensor something which satisfies $T(a X+bX', Y,Z)=aT(X, Y, Z)+bT(X',T,Z)$ for $a,b$ to functions (and similarly for, $Y,Z$).
Then, you choose a complex chart $(z_1,..., z_n)$, and you check it for the fields $\partial \over {\partial z_i} $, ${\partial \over {\partial z_i }} = i {\partial \over {\partial z_i}}$. This is enough, because these fields generate all vector fields.
Note that these fields commute, and furthermore that $i {\partial \over {\partial z_i}}$ is the local expression for $J\partial \over {\partial z_i} $. So to prove that $T(X,Y,Z)=0$ it is enough to check it for all fields $X,Y,Z$ such that $X,Y,Z, JX,JY,JZ$ commute