Here's the United size limits:
https://www.united.com/web/en-US/content/travel/animals/in_cabin.aspx
I highlighted the specs:
>>>
A pet traveling in cabin must be carried in an approved hard-sided or soft-sided kennel. The kennel must fit completely under the seat in front of the customer and remain there at all times. The maximum dimensions for
hard-sided kennels are
17.5 inches long x 12 inches wide x 7.5 inches high (44 cm x 30 cm x 19 cm). The recommended maximum dimensions for
soft-sided kennels are
18 inches long x 11 inches wide x 11 inches high (46 cm x 28 cm x 28 cm). Soft-sided pet carriers may exceed these dimensions slightly, as they are collapsible and able to conform to under-seat space without blocking the aisle. With the exception of birds, there may only be one pet per kennel, and
the animal must be able to stand up and turn around comfortably. Two birds may travel in the same kennel.
<<<