Rubik's Cube techniques

Any two corners A and B can be twisted in opposite directions using a commutator. Like the corner 3-cycles, our strategy will be to divide the cube into two independent portions:

Sequence Y will affect only the green layer (the one with the two corners on it), while sequence X will mostly affect the rest of the cube shown in grey, as well as twisting one of the corners in the green layer. The basic strategy is as follows:

1. Construct a sequence X to twist corner A in place. This sequence must not affect anything else on the green layer, except for corner A. However, to make things easier, it is allowed to mess up anything whatsoever in the rest of the cube (shown in grey).
2. Use sequence Y to move corner B to take the place of corner A. This sequence should affect only the green layer.
3. Apply X^-1. This will twist corner B in the reverse direction to A.
4. Apply Y^-1. This will move corners B and A back to their original positions.

Note that it is impossible to twist exactly two corners in the same direction as this violates the laws of the cube.

Below are some examples of corner twists: