Emergence of Non-Abelian Algebraic Structures from Extended Clifford Algebra Representation Spaces

Abstract: This work proves that non-Abelian Lie algebras su (2) and su (3) emerge as algebraic necessities of Clifford algebra in extended representation spaces. The central result is a classification theorem: when the Dirac spinor space is extended to V₄ ⊗ VN with N=2, 3, the commutator constraint with the Clifford algebra generators uniquely yields su (2) and su (3) as the complete non-trivial Lie algebras. The numbers of generators—3 and 8—and the structure constants are determined entirely by the dimension of the extended space via the formula N²−1. The derivation uses only the anticommutation relations of the Clifford algebra and standard Lie algebra theory.