Using coordinates to show left-invariant vector fields form Lie algebras I came across a simple Lie group question while reading Bernard Schutz's Geometrical Methods of Mathematical Physics.  I am brand new to group theory, and this is my first attempt at grasping its physical significance.
He writes (p. 93) that if $\overline{V}$ and $\overline{W}$ are any two left-invariant vector fields on a manifold $G$, then the left translation map $L_g$ (which maps tangent vectors at the identity $e$ to those at $g$) maps $[\overline{V},\overline{W}]$ at $e$ to $[\overline{V},\overline{W}]$ at $g$.  Then he draws a figure of the Lie bracket, and states that the field $[\overline{V},\overline{W}]$ is also left-invariant.  He says, "The reader who is not convinced by the [diagrammatic proof] is invited to use coordinates on $G$ to prove the result".
I am wondering how one uses coordinates to prove this result.  The Lie bracket, of course, can be expressed $$[\overline{V},\overline{W}]=\left(V^i\frac{\partial W^j}{\partial x^i}-W^i\frac{\partial V^j}{\partial x^i}\right)\frac{\partial}{\partial x^j}$$ (summed over the repeated indices).  But how would the operation $L_g$ interact with each of these component terms?  I only know that $L_g(\overline{V}(e))=\overline{V}(g)$, and likewise for $\overline{W}$.
 A: Let $M$ be a smooth manifold and $\varphi: M\to M$ a diffeomorphism. For a vector field $\xi: M\to TM$ one can define a new vector field $\varphi_*\xi$ by the relation:
\begin{equation}
(\varphi_*\xi)|_q = T_p\varphi(\xi|_p)
\end{equation}
where $q = \varphi(p)$. This lead to an automorphism
$\varphi_*: \Gamma(TM)\to \Gamma(TM)$ of the $C^{\infty}(M)$-module of vector fields called the pushforward by $\varphi$.
The heart of your question rely on the following relation:
\begin{equation}
\varphi_*[\xi_1, \xi_2] = [\varphi_*\xi_1, \varphi_*\xi_2]
\end{equation}
Indeed if $M$ is a Lie group $G$, a vector field $\xi$ is left-invariant if and
only if $(L_g)_*\xi = \xi$ for every $g\in G$. So using the previous relation it's obvious that the Lie bracket of two left-invariant vector fields is left-invariant.
Now since this relation works for any diffeomorphisms I don't think it's easier
to prove it in the very particular case $\varphi = L_g$. Also, prooving this relation depends on how you define the Lie bracket and it's usually not done in coordinates. My own opinion is coordinates are not suitable to prove this relation; it would be tedious. For a standard proof I recommand John. M. Lee's Introduction to smooth manifolds.
