...After this I found a put away a piece -turn in position- and put back - the commutator idea- Back than I did not know the term.
This basically principle works on most puzzles, as long as you can change one piece of a face , you can do a 3 cycle, or turn the piece.
I agree with this technique. It is probably the way most people approach new puzzles.
It is also the reason why even very experienced cubers often have big problems solving a Magic or Skewb (Skewb Ultimate

). Magic doesn't allow cycles untill you get it back to rectangular shape and Skewb turns half the puzzle so there is no safe place to "put away a piece".
If you want a real challenge, try the
Nintendo Barrel where every horizontal move turns 10 out of 23 pieces and every vertical move turns 15 out of 23 pieces.