This question is about solving a Rubik's cube using commutators and conjugates - it is a question about the group theory of the Rubik's cube, rather than a "how do I" kind of question.
This video gives a very helpful overview of how to use commutators and conjugates to solve Rubik's cubes and their larger versions.
When it comes to corner moves, the only commutators it describes are 3-cycles and orientation swaps. Both of these now seem very simple and intuitive to me.
However, sometimes my cube gets into a state where all corners are solved apart from two of them on the last face. These may be adjacent or opposite corners. The other six corners are all in the correct positions and oriented correctly, so I just need to swap the remaining two. The edges are solved only in one layer, as I generally solve the corners first.
Swapping two corners doesn't fit the pattern of what can be done with commutators: a 3-cycle swaps the positions of three cubies, and an orientation swap changes the orientation of two, but I need to change the position of two.
I know that algorithms for this exist - one can be found here, for example - and I'm not asking for an algorithm in this question. Rather I would like to understand the following:
Can a corner swap be performed using commutators and conjugates only, or is it a "parity" case that requires a non-commutator move to accomplish?
If it can be done with commutators, is there a nice intuitive way that I could see how to create my own commutator to solve this case? I can now do this easily with 3-cycles and orientation swaps, but not with this kind of 2-cycle.
If it can't, is there some easy principle through which the parity can be removed? I mean something along the lines of the parities in the 4x4 cube (explained towards the end of the video above), where a single quarter-turn removes the parity, putting the cube into a state that can be solved with commutators and conjugates only, although it does take quite a few moves to do so. (I tried making a single quarter turn of my cube, but re-solving it with combinators just put it into the same state, of needing two corners to be swapped.)
(Note: my cube is a 4x4x4, but I'm solving the corners first, so I don't care about moving edges or even centres around. I had assumed the same situation could arise on all cubes if the edges aren't solved, but if this is wrong it would be interesting to know.)