I think the most compelling aspect of the Azula, Mai, Ty Lee trio to me is how they're toxic, but in a self-regulating way. Like, I firmly believe if any two of them tried to be friends alone (as we see them by the end of the show) it wouldn't last, even without the idea of one of them being dropped. They all bring some emotional core to the relationship the other two can't find elsewhere.
Azula gives Mai and Ty Lee acknowledgent, purpose, the idea that they're worthwhile individuals that deserve more than the bare minimum.
Mai is blunt and honest, something that noblewomen, especially pretty noblewomen aren't going to be able to find in any other relationship.
Ty Lee is caring and warm, something both Azula and Mai lack in the rest of their lives.
And it isn't until Zuko comes in that things start to unbalance, during all of season 2, none of the trio get into any kind of fight with each other. They accept the things that happen, even if they suck. Ty Lee doesn't put up a meaningful fight in the circus, and Azula never punishes Mai for disobedience, which is saying something because she was going to throw a ship captain overboard for not following orders. Yet as early as The Beach, we see Azula and Ty Lee fighting, because Zuko's presence emotionally unbalances the group, ultimately leading to the betrayal at the Boiling Rock. Between the three girls, they all desire the same things from each other, they're unified, but you add in Zuko and they begin to conflict because Azula has been conditioned from birth to see Zuko's failures as her successes, Mai, having a crush on Zuko, wants to see him successful, or atleast content, and Ty Lee as far as we know has no strong feelings either way. None of them are in alignment anymore so that toxicity that was quelled before begins to bubble and boil until it eventually pours over and their relationship snapped.
