I'm in the process of learning Lie theory.
Simply connected Lie groups correspond to finite dimensional real Lie algebras. Finite dimensional semisimple real Lie algebras correspond to Satake diagrams.
So, simply connected semisimple Lie groups correspond to Satake diagrams.
A Lie group is a geometric object and is a group. So, the most natural classification task is to classify Lie groups by geometric, topological and abstract group theoretic properties (i.e. "compact", or "simple as an abstract group").
I know, in fact, that such characterization exists: A connected Lie group is semisimple if and only if its radical (largest connected solvable normal subgroup) is trivial.
So, Satake diagrams correspond to simply connected Lie groups with trivial radical.
While this is this is the type of classification I was looking for, the property of being "simply connected with trivial radical" seems a little artbitrary.
This is my question:
What I'm trying to understand is whether the focus on semisimple Lie groups is only because "that's where we have a good understanding", or also because of some natural mathematical domains where "among Lie groups, the semisimple ones are all that matter".