Nice poster, Deke. Any idea what year it is from? ( I guess the AA logo design might also help to be an indicator).
I don't know precisely. We can set an early bound of May 1942 when Mexico entered the war. But as you suggested, the AA logo provides an even better clue. The American Airlines logo seen on the poster was in use from 1945-1962.
So my best guess is that the poster is from 1945, or perhaps the immediate post-war years when it still would have been topical to refer to "three great allies".
I have seen a couple of other AA posters that make direct reference to the war effort and have the earlier logo that was in use from 1934-1945.