I just think of some connection with Cartan type: there exists some kind of Dynkin/Coxeter diagram for CRG in "Broue, Malle, Rouquier - On complex reflection groups and their asociated braid groups". Maybe this would be a way to connect them using Cartan types. Those diagrams are fairly complicated because for CRG, there are not only braid relations of the form (s_i s_j)^m = 1 but also relations involving 3 generators (that's where the circles in the diagrams come from). But I have not seen any later work using those diagrams, but maybe someone else knows of such? As far as I know, those diagrams are nice in the sense that they give not only the CRG but as well a representation of the associated braid group, analogue to the reflection representation for Coxeter groups with relaxing the quadratic relations. But I don't see a use of those diagrams in sage...
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