# Degree of universal cover of simple Lie group

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?

• See this question and the linked questions and references, which together give the result. – Dietrich Burde Jan 17 at 15:41
• Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.) – user320832 Jan 18 at 5:35