i'm a physics student and i'm studying differential geometry:
I have to prove :
$F_* [X,Y] = [F_* X, F_*Y] $
Where F is a diffeomorphism from G to G where G is a lie group. $F_* $ is the differential of F. X and Y are vector field over G.
I have tried to use definition of differential of a function $ F: M -> N$
$ [(F_*)_p (X_p)] (g) := X_p(g \circ F) $ where $g \in C^{\infty} (N) $ Thanks
Indeed, using the definition of pushforward of a vector field by a diffeomorphism, we get: $F_*([X,Y])_q(g)=F_*([X,Y]_{F^{-1}(q)})(g)=[X,Y]_{F^{-1}(q)}(g\circ F)$ $=X_{F^{-1}(q)}(Y(g \circ F))-Y_{F^{-1}(q)}(X(g \circ F))$.
On the other hand:
$[F_*X,F_*Y]_q(g)=F_*(X_{F^{-1}(q)})(F_*Y(g))-F_*(Y_{F^{-1}(q)})(F_*X(g))$
$=X_{F^{-1}(q)}(F_*Y(g)\circ F)-Y_{F^{-1}(q)}(F_*X(g)\circ F)$
$=X_{F^{-1}(q)}(Y(g \circ F))-Y_{F^{-1}(q)}(X(g \circ F))$
In this last step again is used the dfinition of pushforward by $F$.
In case you are not familiar with this concepts (I tried not to ommit any steps and work each side separately so that it was more comprehensible), I recommend you Lee's book, Introduction to Smooth Manifolds, 2013.