... an aircraft under an AOC can only be operated commercially (or for training). If you want to undertake private flights, the aircraft has to be removed from the AOC first.
Where did you hear that, What Next? Every operator I know that manages aircraft flies the owners as a private flight. I have done it myself. I know that the Authority like to see clear delineation between the two, but that it does not (and I think cannot) ban such mixed use. I even know of companies that fly charters then have the same crew fly the same aircraft privately on the same duty, into duty periods beyond AOC FTLs; I think that is rather a poor choice, but I believe it is allowed. Hmmmmm, thinking about it I did that myself on precisely one occasion - first day of my first ever AOC line training.