I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $\mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $\mathfrak{g}$ is finite.
Can anyone provide a proof or reference for any of these equivalent statements?