# Math Help - Lie bracket question

1. ## Lie bracket question

Let $f:M \rightarrow M$ be a smooth diffeomorphism and $X, Y$ be smooth vectorfields on M. Why is it true that
$f_* [X,Y] = [f_*X,f_*Y]$? Here $f_*$ denotes the push-forward of f.

2. Just unwind the definition.
Any smooth function g, and any point p,
$f_*(X)(g)$ is a smooth function and its value at f(p) is $f_*(X)(g)(f(p)) = X(g \circ f)(p)$, that is $(f_*(X)(g)) \circ f = X(g \circ f)$
Now $f_* [X,Y]_{f(p)}(g)=[X,Y]_p(g \circ f )$
$= X_p(Y(g \circ f)) - Y_p(X(g \circ f))$
$= X_p( (f_*(Y)(g)) \circ f ) - Y_p( (f_*(X)(g)) \circ f )$
$= f_*(X)_{f(p)}(f_*(Y)(g)) - f_*(Y)_{f(p)}(f_*(X)(g))$
$= [f_*(X),f_*(Y)]_{f(p)}(g)$