To say Colonel Frank Fitts is a complex guy is to make the understatement of the century. This guy is a study in repression, rage, and self-loathing. In fact, he might be the most layered character in American Beauty… or it might just be that his character is played by the inimitable Chris Cooper.
At first, his defining characteristic seems to be aggressiveness. He's actually like Carolyn in a weird way, but with the violent tendencies turned up to eleven. Like Carolyn, he seems to police his family members to make sure his house is perfectly in order and to his liking. We don't really get to see that in action with his wife, but her completely catatonic and withdrawn behavior makes us think that this is kind of her defense mechanism for his domineering ways.
With Ricky, we definitely get a healthy slice of Frank's violently controlling tendencies. First of all, Frank makes Ricky give him urine samples to make sure he's not doing drugs (of course, Ricky gives him fake pee to avoid issues, since he smokes pot like a chimney). However, that's not enough—he spies on Ricky, and when Ricky does something he doesn't like, he busts into his room, often getting physical.
Nazi Memorabilia Collector
In one incident, Frank discovers that Ricky had shown Jane a piece of Nazi china that his dad, for whatever reason, purchased and kept on display. He's furious that Ricky was messing with his stuff and bursts into his room, gets violent… the whole nine yards. That is, until he realizes that Ricky had been showing the plate to a girlfriend, which apparently makes things better in Frank's eyes.
Trapped in the Closet
That's another thing about Frank—he really approves of heterosexual behavior and he really doesn't like gay people. Or at least, that's what he says. After meeting the Jims next door, for example, he goes on a whole homophobic rant to Ricky:
FRANK: How come these faggots always have to rub it in your face? How can they be so shameless?
That being said, we ultimately learn that we can't take Frank's bluster at face value. He actually makes quite a left turn toward the end of the movie. During his usual spying on Ricky, he learns that Lester and Ricky are up to something together. As the audience, we know it's drugs, but Frank gets the (totally wrong) impression that Ricky has been giving Lester blowjobs.
Frank freaks out, and of course, we think it's because he's upset that (he thinks) Ricky is gay—and when he shows up at Lester's door, we're ready for a brawl. But what happens? Rather than beating him up, Frank actually tries to kiss Lester.
Turns out, Frank was actually in the closet all this time and (like Angela) all that blowhard nonsense was just a way of hiding what he was actually feeling. Unfortunately, his vulnerability doesn't pay off, since Lester isn't actually gay—and the shame of the whole incident seems like it's just too much for Frank to bear. He ends up killing Lester, probably in large part to keep the whole incident quiet so he can go back to pretending to be a whole other person.
Given all that, we'd say he ties with Carolyn for most tragic character, since he gets virtually zero resolution to his problems and zero personal satisfaction.