I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center.
Here, semi-simple means, that the corresponding Lie algebra can be written as a direct sum of simple Lie algebras (having no non-trivial ideals).
Since this is not immediately obvious for me, I wonder if the proof is actually complicated or if I am just too ignorant to see it. If anyone can give an idea of where this fact comes from, I would be thankful.