Usually they've made a couple enemies. Just gang up with their enemies. If it's a group effort, there's only so much management can do at that point (depending on how much leverage they have). It's all dependent on the specific political situation.
This whole image of them being "well-liked" and socially invincible is just a play. Usually this type of person has made a shit ton of enemies. It may or may not be worthwhile depending on how shitty of a situation you're in, how bad the company's stack ranking and layoff quota is, how attached they are to you, etc
If you can get him fired then go for it. Otherwise, you're only harming yourself by letting yourself dwell on it. As for believing him; when my coworker starts rambling about the time the Navy seals came to him for help, I just say "wow man, crazy". I don't believe him, but neither do I confront it. Why bother? I'm not being paid to call out his dumb lies. This is probably what your other coworkers were thinking.
Conversely, actively complaining about it prevents enabling group thing to form. It helps victims identify the situation sooner, to leave of better protect themselves.
If you really wanna out a liar and come out clean on the other end, it's just so much documentation, emotional decision trees, how to phrase & frame & present things, etc. etc. etc.
Often, the compulsive liar will out themselves, create their own demise, or be enabled and further empowered by a toxic system that rewards such types (which if possible, you'd wanna get out of ASAP because prolonged exposure leaves lasting damage to anyone in such systems).
But, you acted as if there was something wrong with those who dont. And as if there was some higher value in enabling and playing along. I disagree with both.