I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your opinion, obviously), since there are a lot of different ideas involved. 



Classification of (compact) Lie groups
First, here's a rough outline of how the classification works:


1. Prove that if G and H are simply connected and have the same Lie algebra, then G and H are isomorphic as Lie groups.
2. Prove that if G is any Lie group, its universal cover $\tilde{G}$ inherits a natural Lie group structure for which G = $\tilde{G}/Z$ where $Z\subseteq Z(\tilde{G})$.


This reduces classification to a) understanding the Lie algebras and b) understanding the centers of simply connected Lie groups.


`3.` Classify (simple) Lie algebras. This is done via root diagrams (Dynkin diagrams).


`4.` For each simply connected compact Lie group, compute its center.


For references, I'd check out Fulton and Harris' book "Representation Theory". I'm not sure if it actually does 4., but that's a fairly easy exercise afterwards (except for perhaps the exceptional groups).


