Under what conditions is the exponential map on a Lie algebra injective?

Question

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map.

In his blog, Terence Tao notes that if a Lie group is not simply-connected, then $\exp$ will not be injective. Conversely, is it true that if a Lie group is simply-connected, then $\exp$ is injective? If not, what is a counter-example?

Answer 1

$SU(2)$ is simply connected, but its exponential map is not injective -- it's a double cover of $SO(3)$, so rotating by $4\pi$ around any axis is the identity.

Answer 2

The blog post already notes that injectivity fails whenever $G$ contains $S^1$ as a (Lie?) subgroup, in particular whenever $G$ is (positive-dimensional and) compact.