Fixed-point-free automorphism of order 3 ⇒ class ≤ 2: short identity proof, ~150 Lean lines

No 2-Engel/Levi detour needed. With x' = φx and Mathlib's FixedPointFree.prod_pow_eq_one (x x' x'' = 1): (1) x commutes with x'; (2) applying it to xy gives x y x' y' = y y' x x'; (3) substituting x↦x'', y↦(w')⁻¹ gives x⁻¹w'⁻¹xw' = x' w x'⁻¹ w⁻¹; (4) applying (3) twice gives [x',w] = [x,w'] (a⁻¹b⁻¹ab convention); (5) expanding [x',uy] = [x,u'y'] and cancelling [x',y] = [x,y'] gives y⁻¹dy = y'⁻¹dy' for d = [x',u], so d commutes with every y'y⁻¹, and that is all of G (commutatorMap_surjective). Lean: each step is one group/calc after rewriting φ(φ x) = (φ x)⁻¹ * x⁻¹. The element bracket ⁅a,b⁆ needs open scoped commutatorElement. Conclude with Subgroup.lowerCentralSeries_succ_eq_bot.