Linearization of graphic toposes via Coxeter groups

In an associative algebra over a field K of characteristic not 2, those idempotent elements a, for which the inner derivation [−,a] is also idempotent, form a monoid M satisfying the graphic identity aba=ab. In case K has three elements and M is such a graphic monoid, then the category of K-vector spaces in the topos of M-sets is a full exact subcategory of the vector spaces in the Boolean topos of G-sets, where G is a crystallographic Coxeter group which measures equality of levels in the category of M-sets.