TITLE: How do I stop worrying about root systems and decomposition theorems (for reductive groups)?
QUESTION [18 upvotes]: I apologize for this being a very very vague question. 
Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall call these $\textbf {LAG} $objects for short). My "psychological" response to these objects seem to be--they are really "ugly" and unnatural, where do they come from?? Of course, we can say, OK, let's work out the theory of $sl_2$, then everything is just "natural" generalization of easier cases. But this explanation just doesn't satisfy me.
I've been worrying about such things for really quite a long time. And probably the real question is, do these root systems have some (really nice, really simple) geometrical interpretation? For example, is it possible that they are somehow related to some nice topological space, some sheaf, some cohomology, etc...? And do they have some nice counterparts in other branches of mathematics? (it seems to me they only appear in places like LAG objects), do they show up in some other (probably surprising) places?
Actually, as we know, study of LAG objects themselves are closely related to their representation theories. And their representation theory are also obejects that seem hard to visualize in some "easy" geometrical way. I know we can probably put these things as a category and do some abstract algebraic geomtry with them. But still, we cannot avoid using these ad-hoc techniques with them. (like root systems, Dynkin diagrams, etc, etc)
Perhaps I should stop here. In any case, I hope this question will not be closed as a spam. And hope some one can understand some of my confusion and shed some light (geometry) on it.
Thank you.

REPLY [3 votes]: I had a problem which may be similar when I encountered root systems for the first time.  In retrospect, I think that the problem was that I was reading about specific realizations of root systems (eg, the $A_n$ roots embedded into $\mathbb R^{n+1}$ as $e_i-e_j$ , etc.).  Those specific realizations looked very ad hoc to me.  The point, I would say, is that, while it's good to spend some time with some of those specific realizations, you should get used to thinking about root systems in a uniform way, at which point their power becomes pretty evident.  (But maybe this is really a different problem from yours.)