A₁ as the Universal Lie-Algebraic Skeleton of Qubit Symmetry — E8 Intelligence Research

FINDING: The A₁ root system (sl(2,C)) is the fundamental Lie-algebraic skeleton of qubit Hilbert space, with its principal elements and embeddings into sl(m,K) revealing universal structural constraints; Malcev algebras and Nichols braided Lie algebras extend this to non-associative and braided quantum symmetries. | MATH: A₁ ≅ sl(2,C) has root system {±α}, Cartan matrix [2], Weyl group ℤ₂; spin-1/2 Hilbert space ℂ² is the fundamental irrep with Pauli generators σ_x, σ_y, σ_z satisfying [σ_i,σ_j]=2iε_ijk σ_k. Frobenius Lie algebra principal element: e = Σ h_i (Cartan subalgebra generator) with semisimple part having eigenvalues related to root values; embedding theorem: any Lie algebra with left-symmetric (pre-Lie) structure embeds into sl(m,K) — for A₁, m=2 gives the qubit. Malcev algebra correspondence: a Lie algebra L(M) uniquely determined by Malcev algebra M, with anti-commutator [x,y] and Jacobiator proportional to associator — for A₁, the Jacobi identity holds trivially. Nichols Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com