Lie Algebra Commutator

SO(3) has generators Jₓ, Jᵧ, J_z satisfying [Jₓ,Jᵧ] = iJ_z. The commutator measures the failure of rotations to commute. Geometrically: apply ε-rotation A then B, then reverse A then B — the result is an ε²-rotation about the third axis.

SO(3) structure constants:
[Jₓ, Jᵧ] = iJ_z
[Jᵧ, J_z] = iJₓ
[J_z, Jₓ] = iJᵧ

// Commutator sequence:
1. R_A(ε) — rotate by ε around A
2. R_B(ε) — rotate by ε around B
3. R_A(-ε) — undo A
4. R_B(-ε) — undo B

Result ≈ R_C(ε²) where C=[A,B]
= ?
Rotation Matrices
Key insight:
Individual rotations: O(ε)
Commutator error: O(ε²)

This ε² term IS the Lie bracket — the infinitesimal generator of the third rotation axis.

SU(2) double cover:
Same algebra, but spin-½ needs 4π for full rotation!