State-spaces of one-, two-, and three-qubit systems can be described in terms of the complex, quaternionic, and octonionic Hopf fibration respectively. These can in turn be connected to the Lorentz groups in 3+1-, 5+1-, and 9+1-dimensional spacetimes via the automorphisms of the fiber bundle base. In the latter two cases, singling out a preferred complex direction breaks each of these down to 3+1 dimensions. Furthermore, requiring that direction to be invariant under the action of the equivariance group of the Hopf map leaves a residual SU(2)×U(1)/Z2 symmetry in the quaternionic case, and SU(3)×SU(2)×U(1)/Z6, the Standard Model gauge group, in the octonionic one. We speculate on possible implications for model building and exhibit a construction of the degrees of freedom of a single generation of Standard Model fermions.