No, you don't!
From the terms and conditions of American Airlines AAdvantage program:
You may accrue mileage only for purchased, eligible, published-fare
tickets on qualifying routes used in accordance with all applicable
conditions of carriage, tariffs, rules and terms of ticketing and
travel. AAdvantage mileage accrual eligibility on airline participant
routes is subject to change without notice.
and even more clear:
Certain airline tickets are not eligible for earning mileage credit.
These include, without limitation, the following: all tickets issued
as AAdvantage awards or other free ticket promotions including free or
reduced rate tickets;...
As you don't get new mileage on AA, you most certainly also don't get any on any other airline.