A₁ Root System as the Universal Algebraic Foundation of Qubit Hilbert Space — E8 Intelligence Research

FINDING: The A₁ root system (rank-1 simple Lie algebra, sl(2,K)) is the foundational algebraic structure for spin-1/2 qubit Hilbert space, with its principal elements and embeddings into sl(m,K) revealing universal Frobenius/Lie-algebraic constraints. | MATH: A₁ ≅ sl(2,K) ≅ su(2)⊗ℂ; root system Φ = {±α}, α² = 2; Cartan matrix [2]; Weyl group ℤ₂; fundamental rep: 2-dimensional Hilbert space ℂ² (qubit); Pauli matrices σ_x, σ_y, σ_z generate su(2) with [σ_i, σ_j] = 2iε_ijk σ_k; Casimir invariant C₂ = (1/2)σ·σ = (3/4)I₂; principal nilpotent element e (Gerstenhaber–Giaquinto) satisfies ad(e)² ≠ 0, ad(e)³ = 0 in sl(2); embedding theorem: any Lie algebra with left-symmetric (pre-Lie) structure embeds into sl(m,K) — for A₁, m=2 minimal. | CONNECTION: A₁ root system is the crystallographic root system of type A₁ — the only rank-1 case, with root length ratio 1:1 (no longer/shorter roots), corresponding to the single simple root α with Cartan integer 2. The Weyl group order |W| = 2 = 2¹, matchin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com