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Consider the following properties of a compact connected Lie group $G$:

(a) $G$ is semi-simple,

(b) $G$ has a finite fundamental group.

The well known Weyl's theorem states that (a) implies (b).

There are results in the structure theory of compact groups that lead me to believe that the converse implication is also true. If this is the case, can you provide me with a reference? If the implication fails to hold and I am mistaken, can you please suggest a counterexample?

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This is in (e.g.) Bourbaki, Lie groups and Lie algebras, Chap. 9, §1, no. 4, Cor. 4.

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