>
> I will push this patch, and as well a first implementation of the rest
>> later today (though I find the organization still obscure, maybe we
>> can spend some time on clarifying things here).
>>
>
> Okay, a first draft seems to be working.
>

I forgot to give a single example: The first CoxeterGroup is my
implementation using chevie, everything after is purely in Sage.

sage: W  = CoxeterGroup(['H',3])
sage: M = W.cartan_matrix()
sage: M
[              2 E(5)^2 + E(5)^3               0]
[E(5)^2 + E(5)^3               2              -1]
[              0              -1               2]
sage: DD = DynkinDiagram(M)
sage: RS = DD.root_system()
sage: SP = RS.root_space()
sage: SP.weyl_group().cardinality()
120
sage: len(SP.roots())
30

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