If $\displaystyle X$,$\displaystyle Y$ are vector fields on a manifold $\displaystyle M$ and $\displaystyle f$,$\displaystyle g:{M}\rightarrow {R}$ differentiable functions, show that
$\displaystyle [fX,gY]=fg[X,Y]+f(Xg)Y-g(Yf)X$

Also if $\displaystyle X=x_{2}\frac{\partial}{\partial{x_{1}}}-x_{2}\frac{\partial}{\partial{x_{2}}}$, $\displaystyle Y=x_{3}\frac{\partial}{\partial{x_{2}}}-x_{2}\frac{\partial}{\partial{x_{3}}}$, $\displaystyle Z=\frac{\partial}{\partial{x_{1}}}+\frac{\partial} {\partial{x_{2}}}+\frac{\partial}{\partial{x_{3}}}$ are vector fields on $\displaystyle R^3$, calculate the components of $\displaystyle [X,Y]$,$\displaystyle [X,Z]$,$\displaystyle [Y,Z]$.

Finally, let $\displaystyle (U,f)$ be a map of the $\displaystyle m$-dimensional manifold $\displaystyle M$ and $\displaystyle X$,$\displaystyle Y$ be two dimensional fields onto $\displaystyle U$. If $\displaystyle \frac{\partial}{\partial{x_{i}}} (i=1,...,m)$ are the basic vector fields of $\displaystyle (U,f)$ and $\displaystyle X=\sum_{i=1}^{m}a_{i}\frac{\partial}{\partial{x_{i }}}$,$\displaystyle Y=\sum_{i=1}^{m}b_{i}\frac{\partial}{\partial{x_{i }}}$ where $\displaystyle a_{i}$,$\displaystyle b_{i}
\in C^{oo}(U,R)$ show that $\displaystyle [X,Y]=\sum_{i=1}^{m}(\sum_{j=1}^{m}(a_{j}\frac{\partial {b_{i}}}{\partial{x_{j}}}-b_{j}\frac{\partial{a_{i}}}{\partial{x_{j}}})\frac {\partial}{\partial{x_{i}}}$.

($\displaystyle [,]$ denotes the Lie bracket on vector fields) I am not familiar with the Lie bracket at all, so I would be happy if someone helped!