Lie Groups & SO(3) Rotations

The Lie group of 3D rotations and its Lie algebra so(3)

R(θ,n̂) = exp(θ·[n̂×]) = I + sin(θ)[n̂×] + (1−cos θ)[n̂×]²   (Rodrigues' formula)
SO(3) is the Lie group of 3D rotations: 3×3 matrices with RᵀR=I and det=1. Its Lie algebra so(3) consists of 3×3 antisymmetric matrices — the generators Jₓ, Jᵧ, J_z. The exponential map exp: so(3) → SO(3) gives Rodrigues' formula. SO(3) is non-abelian: R₁R₂ ≠ R₂R₁. The double cover SU(2) → SO(3) explains spin-1/2 particles requiring 4π rotations.