score:3
You can take up to three bags, each weighing to 32 kg, for free on every flight on this journey.
AA allows business class customers to take up to three bags for free, at 32 kg each.
1st, 2nd, and 3rd bag fees are waived for business class customers: https://www.aa.com/i18n/travelInformation/baggage/checked-baggage.jsp
Overweight bag fee is waived for business class customers, up to 32 kg: https://www.aa.com/i18n/travelInformation/baggage/oversize-overweight.jsp
US federal regulations require that you will have a consistent baggage allowance throughout your journey if the journey starts or ends in the United States. Therefore this allowance will be applied throughout your journey including onto Qatar Airways.
In practise, AA takes the most generous allowance that would apply at any point, and applies that throughout the ticket.
If you hold Emerald status with any oneworld airline, you will also get a fourth bag for free.
If any other carrier on this journey would give you an even more generous allowance, then the more generous allowance will apply.