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Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4,as well as those of E_8, have been used extensively in the literature. Thepresent paper analyses such Coxeter groups in the Clifford Geometric Algebraframework, which affords a simple way of performing reflections and rotationswhilst exposing more clearly the underlying geometry. The Clifford approachshows that the quaternionic representations in fact have very simple geometricinterpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodgedualised root vectors, which determine the reflection planes of the Coxetergroups. Two successive reflections result in a rotation, described by thegeometric product of the two reflection vectors, giving a Clifford spinor. Thespinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a newsimple construction of binary polyhedral groups. These in turn generate thegroups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely usedquaternionic representations are shown to be spinors in disguise. Therefore,the Clifford geometric product in fact induces the rank-4 groups from therank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptionalstructures, which our study sheds new light on.

Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups