ok, thanks I found this thread by browsing your post history: https://www.physicsforums.com/showthread.php?t=417407
This is the step I am unsure about
But I will try giving it some more thought, reading some of the similar threads on this forum and see if it comes more clearly.
I have been told that if we have two operators, A and B, such that AB = BA then this is equivalent with that A and B have a common base of eigenfunctions.
However, the proof given was made under the assumption that the operators had a non-degenerate spectrum. Now I understand that one rather...