Malcev Algebras, Unique Lie Correspondence, and su(2) Pauli Realization — E8 Intelligence Research

FINDING: Malcev algebras (non-associative, satisfying the Malcev identity) admit a unique corresponding Lie algebra, and su(2) with Pauli basis provides the canonical physical realization of the associator/Jacobiator structure. | MATH: Pauli basis: σ₁σ₂ = iσ₃ (cyclic), {σᵢ,σⱼ} = 2δᵢⱼI, [σᵢ,σⱼ] = 2iεᵢⱼₖσₖ. Associator: (x,y,z) = (xy)z − x(yz). Jacobiator: J(x,y,z) = [[x,y],z] + [[y,z],x] + [[z,x],y]. Malcev identity: (xy)(xz) = ((xy)x)z + ((yz)x)x + ((zx)x)y. Lie algebra correspondence: L(M) = M ⊕ M with bracket [(a,b),(c,d)] = ([a,c]−[b,d], [a,d]−[b,c]). | CONNECTION: su(2) ≅ quaternions (unit) ≅ S³; Pauli matrices span a 3D real vector space with structure constants εᵢⱼₖ (Levi-Civita). Root system A₁: two roots ±α, Weyl group order 2. Killing form on su(2): B(X,Y) = 4·tr(XY), signature (3,0) negative-definite. Casimir: C = ¼(σ₁²+σ₂²+σ₃²) = ¾I for spin-½. Ratio of spin eigenvalues: m ∈ {−½, +½}, spacing ratio 1:1 (degenerate ladder). No φ-ratios appear intrinsically in su(2); however, t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com