I am reading the representation book by Fulton & Harris. In page 119, just below Proposition 8.41, the authors say that "From what we have seen, this Lie group is unique if we require it to be simply connected, and then all others are obtained by dividing this simply connected model by a discrete group of its center". I cannot understand this sentence well.
In Lecture 7, the concept of isogeny was introduced, but at that time nothing was said about the Lie algebra homomorphism induced by the quotient map. So my question is: if two Lie groups $G$ and $H$ are isogenous, how to show that their Lie algebras are isomorphic?
I think the exponential map may be used, but I just have no idea where to start.
I have to admit that I feel quite confused when reading Lectures 7&8, which is partly because of my limited knowledge about geometry. So any nice references for the two lectures are welcomed.