You are given three smooth vector fields on a differentiable manifold $M$. Take the Lie bracket:
$[X,Y]f=X(Yf)-Y(Xf)$
My question is what is the law of multiplication between $X$ and $Y$? Composition of functions?
If the vector field can be written as $X=X^\mu\partial_\mu$, then the above expression becomes:
$[X,Y]f=X^\mu(\partial_\mu Y^\nu)\partial_\nu f +X^\mu Y^\mu \partial_\mu \partial_\nu f-Y^\nu(\partial_\nu X^\mu)\partial_\mu f - Y^\nu X^\mu \partial_\nu \partial_\mu f$
Why is it allowed to change the order of $Y^\nu$ and $X^\mu$ and of the derivatives?
Recall that vector fields are given by derivations on the space of smooth functions. That is, they are maps $$ D_X : C^\infty(M) \to C^\infty(M) $$ In that case, it is really just the composition $$ C^\infty(M) \xrightarrow{X} C^\infty(M) \xrightarrow{Y} C^\infty(M) $$ In coordinates, of course, this is just taking successive derivatives.