TITLE: Every 1- or 2-dimensional compact, connected Lie group is abelian
QUESTION [5 upvotes]: I know that in the 1-dimensional case the conclusion follows immediately from the fact that there exists a maximal torus (which then has to be the group itself).
In the 2-dimensional case, we also know that there exists a maximal torus, and so the point is to show that it must have dimension 2 (or equivalently, that the dimension cannot be 1). I've been trying to prove this for quite a while, but unfortunately I'm stuck and really don't know what to do.
I understand that this is probably an easy question, but my background is in physics and I'm a beginner in these more abstract topics of mathematics; as such, any help would be greatly appreciated.

REPLY [5 votes]: The Lie algebra $\mathfrak g$ of a connected compact Lie group $G$ is unimodular: this means that for all $x\in\mathfrak g$ the map $\operatorname{ad}(x):\mathfrak g\to\mathfrak g$ has zero trace. 
Now let $G$ be a compact Lie group of dimension two. Its Lie algebra, by the result above, is unimodular. Now, there is exactly one Lie algebras of dimension two which is unimodular, and it is the abelian one. This means that the universal covering $\tilde G$ of $G$ is an abelian Lie group (indeed, this group $\tilde G$ is a simply connected Lie group with the same Lie algebra as that of the simply connected Lie group $\mathbb R^n$, so the groups $\tilde G$ and $\mathbb R^n$ are in fact isomorphic: this is the correspondence at work), and therefore $G$ is also abelian.