Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction

In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. In a Clifford algebra framework, an even number of successive Coxeter reflections yields - via the Cartan-Dieudonne theorem - spinors that describe rotations. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4. For the corresponding case in two dimensions, the groups I_2(n) are shown to be self-dual.