Lie algebra of $\mathbb{R}^{n}$ Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the differentiable group operation $+$). Is it right that the lie algebra of this lie group is $\mathfrak g = \mathbb R^n$ with the exponential map $\exp : \mathfrak g \rightarrow G: x \mapsto x$? What is the lie bracket of $\mathfrak g$? Is $[x,y]$ always zero, because $+$ is commutative?

Yes and yes. (Exercise: exhibit $\mathbb{R}^n$ as a Lie subgroup of $\text{GL}_{n+1}$. If you get stuck, try the case $n = 1$ first.)  
