  In this paper, the basis states of the minimal left ideals of the complex
Clifford algebra $C\ell(8)$ are shown to contain three generations of Standard
Model fermion states, with full Lorentzian, right and left chiral, weak
isospin, spin, and electrocolor degrees of freedom. The left adjoint action
algebra of $C\ell(8)\cong\mathbb{C}(16)$ on its minimal left ideals contains
the Dirac algebra, weak isopin and spin transformations. The right adjoint
action algebra on the other hand encodes the electrocolor symmetries. These
results extend earlier work in the literature that shows that the eight minimal
left ideals of $\mathbb{C}(8)\cong C\ell(6)$ contain the quark and lepton
states of one generation of fixed spin. Including spin degrees of freedom
extends $C\ell(6)$ to $C\ell(8)$, which unlike $C\ell(6)$ admits a triality
automorphism. It is this triality that underlies the extension from a single
generation of fermions to exactly three generations.
