Originally Posted by orbitmic

In my view the easiest option is to go to the BA website, go to MMB, ask to select/change seats, select the AA operated flights. BA will then tell you it needs to take you to the AA website and asks for your authorisation, but in doing so, at the bottom of the screen, they will tell you what the AA booking reference is and you can just write it down.

Of course, you can also call BA or call AA. Both will be able to tell you based on the info you have (BA reference, passenger name and in the case of AA flight)