Let $G$ be a connected Lie group and let $\mathfrak{g}$ be its Lie algebra. I am trying to prove that $G$ is abelian if and only if $\mathfrak{g}$ is abelian. My thoughts are as follows:
I think it's enough to show that for any $X,Y\in\mathfrak{g}$, there holds $[X,Y]=0$ iff \begin{equation} exp(tX)exp(sY)=exp(sY)exp(tX),\quad\text{for all $s,t\in\mathbb{R}$}, \end{equation} and prove that $G$ is abelian if and only if also the above equation holds.
My question is how to prove these two equivalent propositions? For the first one, I tried to use the uniqueness of the integral curve and the definition of Lie derivative, but I don't know how to computation the "derivative". For the second one, I was confused about that if the equation holds, how to prove $G$ is abelian? I'm not sure whether for any $g\in G$, there must exists an integral $\gamma$ such that $\gamma(s)=g$.
Thanks for any help!
