American Airlines (NASDAQ:AAL) has announced a significant change in its pet policy, allowing passengers to bring a full-size carry-on along with their pet carrier. What Happened: The airline’s new policy states that passengers can bring a pet carrier as a carry-on if they pay the carry-on pet fee and their pet remains in the carrier under the seat in front of them for the entire flight. They can also bring one additional item on board with their pet, such as a purse, small handbag, or a carry-on bag that meets the airline’s requirements, USA Today reported on Monday. “On flights with American…