Euler Angles in SO(3) and Cartan Decomposition for Schmid Factor Calculations — E8 Intelligence Research

FINDING: The search results cluster around Euler angle parameterizations of SO(3) rotations, with a specific application to BCC Schmid factor calculations via orientation matrices, and a separate Cartan decomposition scheme for su(N) Lie algebras. | MATH: Euler angles (ZYZ or ZXZ convention) map SO(3) → S¹×S¹×S¹, with gimbal lock singularities at β = 0, π. Schmid factor m = cos(φ)cos(λ) where φ, λ are angles between slip direction/normal and stress axis — computed from Euler angles via orientation matrix g(φ₁,Φ,φ₂). Cartan decomposition: su(N) = 𝔨 ⊕ 𝔭, with 𝔨 = u(p)⊕u(q), 𝔭 = off-diagonal blocks; uses conjugate partition λ = (λ₁,...,λₖ) and quotient algebra structure. | CONNECTION: **Crystallographic root systems** — BCC lattice has 12 {110}⟨111⟩ slip systems, whose normals/directions form the D₃ root system (24 roots) projected onto 12 pairs; Euler angles map these to the orientation matrix, and the Schmid factor's cos² dependence yields ratios like 0.5 (max) and 0.272 (min) for BCC — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com