Conceptual reason why Coxeter groups are never simple Is there a conceptual reason why (non-abelian) Coxeter groups are never simple?
For example is there some obvious normal subgroup that can be defined? Or perhaps it is for some reason clear that the commutator subgroup of a Coxeter group is always proper?
I was looking through the list of Coxeter groups and noticed that a lot of them are pretty close to simple, but not quite. For example the Coxeter groups of type $ A_n $ are $ S_{n+1} $ which has the simple index $ 2 $ alternating subgroup $ Alt_{n+1} $ for $ n \geq 4 $.
Coxeter groups of type $ E_6, E_7, E_8, H_3 $ also have index 2 simple subgroups: $ PSp_4(3), Sp_6(2), GO_8^+(2),Alt_5 $ respectively. The Coxeter groups of type $ B_n=C_n $ and type $ D_n $ have a single non-abelian simple composition factor: $ Alt_n $, but these groups are not simple.
The Coxeter groups of type $ F_4 $ and $ I_2(n) $ are solvable. Finally, the Coxeter group of type $ H_4 $ has an index $ 4 $ subgroup $ Alt_5 \times Alt_5 $. So by exhaustion no Coxeter group is simple. I was wondering if there is a nice conceptual reason for that, especially since some of them get so close to being simple (like $ A_n, E_6,E_7,E_8, H_3 $), but aren't quite.

Yes there is an obvious normal subgroup of index $2$ consisting of elements defined by words of even length in the generators.
The same applies to any group defined by a presentation in which all defining relators have even length.
