verify these three differential operators form a lie algebra

Question

I am self studying Lie algebra topic at the basic level. however it is not really clear for me what do a pair of brackets mean exactly?I have collected a bunch of criteria for two operators to form a lie algebra. however I can not find a clear example how to apply them. I appreciate a step by step help
for example do $D_x$ and $xD_x$ and $x^2D_x$ form a lie algebra?

Answer 1

$[D_x,xD_x](f)=D_x(xD_x(f))-xD_x(D_x(f))=D_x(xf')-xD_x(f')=f'+xf"-xf"=f'=D_x(f)$

Compute $[D_x,x^2D_x], [xD_x,x^2D_x]$ and show that they are elements of the vector space generated by $D_x, xD_x$ and $x^2D_x$.