Originally Posted by ISTFlyer

If you would like to book on AA stock, you could use the Powertools extension on ITA Matrix or copy the itinerary to Bookwithmatrix.com.
You could alternatively call AA or BA ( each respectively for their own ticketing stock ) but if I'm not wrong BA is not prioritizing sales calls thus the wait times on the phone might be longer than usual.

Bookwithmatrix.com can also copy the same itinerary to Priceline and some other OTA's but when booking with them, be aware that for changes, you would need to deal with the agency directly and tickets might be issued in Iberia or Finnair ticketing stocks which don't offer that much flexibility compared to American and British.

You could always use a travel agent but they are likely to have a small service fee.

Also a slight note, if booking on AA ticketing stock, be aware that AA would not budge to refund a ticket if there is a cancellation and a legitimate option ( could be most direct ) is available within -4/+4 hours unless you file a complaint to a regulator ( US DoT, EC261/2004, etc... ) thus if booking with AA, also consider that.