Check the UA rules as well as those of any other carrier you're using to avoid the possibility of an unpleasant surprise and please don't attempt to cheat on the birthdate. Some airlines have limits on the number of children of certain ages versus the number of adults (usually defined as over 18 or the official parent or guardian) before UM fees become mandatory or the group can't travel. You should be OK, although on most carriers, the 12 year old can't supervise the 4 year old, and the adult cannot be seated in a different cabin.

It would be less stressful for you and everyone else if the lap baby had its own seat.