  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. The
present paper analyses such Coxeter groups in the Clifford Geometric Algebra
framework, which affords a simple way of performing reflections and rotations
whilst exposing more clearly the underlying geometry. The Clifford approach
shows that the quaternionic representations in fact have very simple geometric
interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3
and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge
dualised root vectors, which determine the reflection planes of the Coxeter
groups. Two successive reflections result in a rotation, described by the
geometric product of the two reflection vectors, giving a Clifford spinor. The
spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new
simple construction of binary polyhedral groups. These in turn generate the
groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely used
quaternionic representations are shown to be spinors in disguise. Therefore,
the Clifford geometric product in fact induces the rank-4 groups from the
rank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional
structures, which our study sheds new light on.
