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Jun 21, 2019 at 8:41 comment added Doriano Brogioli Thanks for the editing and for the comment. I would like to drive the attention of the readers to my answer, which is the final outcome of the discussion. That's why I "accepted" it. If there is a consensus against this, or an alternative suggestion, then I will change. Basically, the question asks what happens if two Hamiltonians are in involution, limiting to quadratic forms defined by $A$ and $B$ and to linear canonical transformations $M$. Thus requiring positive definitness is a more natural condition for me. But I acknowledge the contribution of the other participants.
Jun 20, 2019 at 21:51 history edited LSpice CC BY-SA 4.0
Added links to referenced answers and comments
Jun 20, 2019 at 21:46 comment added LSpice I'm not sure of the consensus, but I think it is better form to accept someone else's answer that inspired your own.
Jun 20, 2019 at 12:06 history edited Doriano Brogioli CC BY-SA 4.0
added 20 characters in body
Jun 19, 2019 at 20:31 history edited Doriano Brogioli CC BY-SA 4.0
added 39 characters in body
Jun 19, 2019 at 20:14 history edited Doriano Brogioli CC BY-SA 4.0
Exchanged A and B
Jun 19, 2019 at 20:11 vote accept Doriano Brogioli
Sep 2, 2019 at 11:42
Jun 19, 2019 at 16:15 history answered Doriano Brogioli CC BY-SA 4.0