  In this paper, we show that via a novel construction every rank-3 root system
induces a root system of rank 4. Via the Cartan-Dieudonn\'e theorem, an even
number of successive Coxeter reflections yields rotations that in a Clifford
algebra framework are described by spinors. 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, we show
that these polytopes have to be root systems, thereby inducing Coxeter groups
of rank 4, and that their automorphism groups include two factors of the
respective discrete spinor groups trivially acting on the left and on the right
by spinor multiplication. Special cases of this general theorem include the
exceptional 4D groups $D_4$, $F_4$ and $H_4$, which therefore opens up a new
understanding of applications of these structures in terms of spinorial
geometry. In particular, 4D groups are ubiquitous in high energy physics. For
the corresponding case in two dimensions, the groups $I_2(n)$ are shown to be
self-dual, whilst via a similar construction in terms of octonions each rank-3
root system induces a root system in dimension 8; this root system is in fact
the direct sum of two copies of the corresponding induced 4D root system.
