Lie bracket on vector fields I started reading about Lie derivative on vector fields and its properties, found an exercise, but i have doubts about my solution.
Given are two vector fields $X_{1}=\frac{\partial }{\partial x_{1}} + x_{2}\frac{\partial }{\partial x_{3}} + x_{2}^{2}\frac{\partial }{\partial x_{4}}$ and $X_{2}=\frac{\partial }{\partial x_{2}}$. Calculate $\left [ X_{1},\left [ X_{1}, X_{2} \right ] \right ]$.
OK, i start with $\left [ X_{1},\left [ X_{1}, X_{2} \right ] \right ] = \left [ X_{1}, \left [ X_{1}X_{2}-X_{2} X_{1}\right ] \right ] = X_{1}X_{1}X_{2} - X_{1}X_{2}X_{1} - X_{1}X_{2}X_{1} + X_{2}X_{1}X_{1} = \left [ X_{1}, X_{1}X_{2} \right ]- \left [ X_{1}, X_{2}X_{1} \right ]$
The book with the answers says that the result of $\left [ X_{1},\left [ X_{1}, X_{2} \right ] \right ]$ should be $0$.
Is my idea correct? Can we replace $X_{1}X_{2}$ with $X_{2}X_{1}$? I am not sure about this. The other thing that bothers me is that i didn't use the definitions of $X_{1}$ and $X_{2}$ the way they were given in the exercise. Can somebody help me? 
Thank you very much

Your calculation is correct so far, but to proceed further you need to substitute in the specific $X_1$ and $X_2$ that were given.  So far your calculation works for any $X_1$ and $X_2$.  Remember that $\partial/\partial x_i$ and $\partial/\partial x_j$ commute.
