Proving a 1 Dimensional Lie Algebra is Abelian

Question

This question is pretty basic but I am but a simple physics postgrad studying Lie Algebras for the first time and want to check my understanding...

I am interested in showing that a rank one lie algebra is abelian.

$$ [x,x] = 0 \quad\forall x \in \mathfrak{g} $$

If I have proved this successfully am I also correct that if the Lie Algebra is abelian then the Lie Group of the same dimension is also abelian.

I thought that seemed a little basic and I wonder if I have missed anything else?

Answer 1

Let $\{e_1\}$ be a basis of your $1$-dimensional Lie algebra $L$ over a field $K$. The Lie bracket $[x,y]$ is determined by the brackets of the basis elements. So the only bracket to consider is $[e_1,e_1]$, which is zero by the definition of a Lie algebra.