I just have a practical question. I search at ITAsoft and like one of
the itineraries/price they come up with.

I just did this. You are correct, ITA doesnt sell tickets...they allow you
to use their engine for free...they are actually in the business of selling
their software to the airlines.

I like a piece of paper in my hand, so I printed my ita-generated itinerary
(that was not offered on delta.com). Then I called Delta and had the agent
book the itinerary that I had in my hand. It worked...no problem.

In my case, I told the agent that the itinerary wasn't offered online, what
the price was, I may have even given him some of the flight/code numbers
from the ITA data....and that's what I paid.

So call AA and see what happens...we want a full report!

However, they don't sell the
ticket nor give me any link. Now what do I do to buy that itinerary,
online if possible?

It is possible that there may be a small extra fee for booking with a live
person.