I've been reading the "geometry of the classical groups" (by Taylor) and got to the part about the BN pair and the Weyl group. as I understand this structure suppose to give the groups something similar to the triangular matrices (B) , and permutations matrices (W), but a part from this, I'm pretty lost.

I can follow the proofs (most of the time) and understand them but I cannot see the "big picture" - meaning that I can't understand why he chose to prove the way he did, or why the result of the theorems are so important, etc.

For example, the double cosets of (B,B) is used a lot there - what exactly do they mean? I mean that if we take the stabilizer of some group action, then each left coset correspond to an element in the orbit, so what does the double coset here means?

any way, if someone can give me a bit of intuition about this subject, or point me to a good site\book I'll be gratefull.

one more thing, I don't know what are lie groups (for now)... and so I cannot use most of the books that deals with BN pairs