Recently I flew AA 17 (JFK-SFO) in F. Two people flying, and though one had become displaced from the originally booked seat, both ultimately settled in their bulkhead Flagship suites. Each had a soft-sided carrier containing a cat.

They both required to place their soft-sided carrier into the "Cuddle Class" shelf space during taxi, takeoff and landing. The rest of the time, the passengers were allowed to keep the kennels in their seat area (generally under the foot shelf).

As can be seen in the illustration, the space for keeping the animal during taxi, takeoff and landing is in fact the lower portion of a closet with a ventilation grille in the door. There are actually two of these, immediately between bulkhead and first suites.