The sexy move as a commutator
[R, U]RUR'U'
Four moves you have done thousands of times. A is R, B is U: turn R, turn U, undo the R, undo the U.
11 cases
Commutators
Build your own algorithms: swap exactly three pieces and leave everything else alone.
A commutator is the pattern [A, B] = A B A′ B′, and it is why most last-layer algorithms look the way they do. Understanding it turns memorisation into construction: you can work out an algorithm for a case you have never seen, and it is the foundation of blindfolded solving. This set is a teaching sequence rather than a list to drill.
From expert
The same commutator backwards
[U, R]URU'R'
The sexy move with the two halves swapped over.
Six sexy moves
(R U R′ U′)6RUR'U'RUR'U'RUR'U'RUR'U'RUR'U'RUR'U'
The sexy move done six times over. The cube comes back solved.
A conjugate: setup, act, undo
[F : [R, U]]FRUR'U'F'
The sexy move with an F in front of it and an F prime behind. You know this one as an OLL, and as the cheapest CMLL case in the library.
A pure edge three-cycle
[M′, U2]M'U2MU2
Four moves. Three edges cycle — the two on the top of the M slice and the front one on the bottom — and nothing else on the cube moves at all.
The edge cycle, moved with a setup
[U : [M′, U2]]UM'U2MU
The same four-move commutator with a U in front. It now cycles the right, left and front-bottom edges instead.
The eight-move corner cycle
[R U R′, D]RUR'DRU'R'D'
Eight moves, three corners cycled — front-right-top, front-right-bottom and front-left-bottom — and not a single edge disturbed.
The corner cycle with a setup move
[U′ : [R U R′, D]]U'RUR'DRU'R'D'U
The eight-move corner commutator with a U prime before it and a U after. A different trio of corners cycles.
Niklas — a corner cycle inside the top layer
NiklasRU'L'UR'U'LU
Eight moves alternating right and left. Three top-layer corners cycle and nothing else moves.
A commutator with a half-turn interchange
[R′ F R, B2]R'FRB2R'F'RB2
Eight moves, three corners cycled, everything else untouched — this time reaching round the back of the cube.
Changing the interchange changes the trio
[R′ D′ R, U2]R'D'RU2R'DRU2
Eight moves again, and again three corners. Compare with the D-interchange version and see which corners moved.