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Division algebras have demonstrated their utility in studying non-associativealgebras and their connection to the Standard Model through complex Cliffordalgebras. This article focuses on exploring the connection between thesecomplex Clifford algebras and their corresponding real Clifford algebrasproviding insight into geometric properties of bivector gauge symmetries. Wefirst generate gauge symmetries in the complex Clifford algebra through ageneral Witt decomposition. Gauge symmetries act as a constraint on theunderlying real Clifford algebra, where they're then translated from theircomplex form to their bivector counterpart. Spin group arguments allow theidentification of bivector structures which preserve the gauge symmetryyielding the corresponding real Clifford algebra. We conclude that StandardModel gauge groups emerge from higher-dimensional Clifford algebras carryingEuclidean signatures, where particle states are recognized as a combination ofbasis elements corresponding to complex Euclidean Clifford algebras.

Probing Clifford Algebras Through Spin Groups: A Standard Model Perspective