Also, is your wife the primary cardholder?

From the T&Cs for everyone's reference:

First checked bag free

For benefit to apply, the CitiBusiness® / AAdvantage® account must be open 7 days prior to air travel, and reservation must include the primary credit cardmember's American Airlines AAdvantage® number 7 days prior to air travel. If your credit card account is closed for any reason, these benefits will be cancelled.

Eligible CitiBusiness® / AAdvantage® primary credit cardmembers may check one bag free of charge when traveling on domestic itineraries marketed and operated by American Airlines, or on itineraries marketed by American Airlines and operated as American Eagle® flights by Envoy Air Inc., Republic Airways Inc., SkyWest Airlines, Inc., Mesa Airlines, Inc., PSA Airlines, Inc., or Piedmont Airlines, Inc. All flights on the itinerary must be domestic flights marketed by American Airlines and operated by American Airlines or American Eagle. This benefit will not be available for travel on codeshare flights booked with an American Airlines flight number but operated by another airline. For the CitiBusiness® / AAdvantage® card, up to four (4) companions traveling with the eligible primary credit cardmember will also get their first checked bag free of charge if they are listed in the same reservation. Benefit does not apply to overweight or oversized bags. This benefit cannot be combined with any existing AAdvantage® elite program benefits, or with First or Business class benefits, including any waiver of baggage charges. Please see aa.com/baggage » for baggage weight and size restrictions. Applicable terms and conditions are subject to change without notice.