I want to prove this claim : If $[X,Y]=0$, then for any tangent field $Z$, $[X,[Y,Z]]=0$
I try to do this by using local coordinate.
$$ X=\sum_{n=1}^{m}a_i\frac{\partial}{\partial x_i}\space\space\space, Y=\sum_{n=1}^{m}b_i\frac{\partial}{\partial x_i}\space\space\space, Z=\sum_{n=1}^{m}c_i\frac{\partial}{\partial x_i}\space\space\space$$
Then, $$ [X,[Y,Z]]=\sum_{t=1}^{m}(\sum_{s,k=1}^{m} a_s\frac{\partial b_k}{\partial x_s}\frac{\partial c_t}{\partial x_k}+ a_s b_k\frac{\partial^2 c_t}{\partial x_s\partial x_k}- a_s\frac{\partial b_t}{\partial x_k}\frac{\partial c_k}{\partial x_s}- a_s\frac{\partial^2 b_t}{\partial x_s\partial x_k}c_k- \frac{\partial a_t}{\partial x_s}b_k\frac{\partial c_s}{\partial x_k}+ \frac{\partial a_t}{\partial x_s}\frac{\partial b_s}{\partial x_k}c_k)\frac{\partial}{\partial x_t} $$
However, I couldn't make any progress.
The result isn't true without additional conditions. For example in $\mathbb{R^3}$, take $X=\frac{\partial}{\partial x}, Y=\frac{\partial}{\partial y}, Z = xy\frac{\partial}{\partial z}$. Then $[X,Y] = 0$, but $[X,[Y,Z]] = [X,x\frac{\partial}{\partial z}] = \frac{\partial}{\partial z}\ne 0$.