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human
[ATLA] How did Ozai manage to get his father to name him heir to be the next Fire Lord?

At the time, wasn't Azulon angry at Ozai and ordered him to kill Zuko? Azulon also died that night from being poisoned and so would have other things on his mind like not dying instead of last minute political changes. Why did everyone believe Azulon legitimately made Ozai the next Fire Lord?
gpt
Ursa specifically used an untraceable poison, so Azulon's death looked entirely natural. It was probably suspicious that Ozai's wife disappeared at the same time Azulon died, but otherwise there was no evidence to point to foul play.

Ozai's word was good enough because Iroh wasn't around to push his claim. And when he did return he just accepted it. 

I also get the sense that Ozai used corruption and bribery to cement his position (since every Fire Nation commander in the show besides Zhao comes off as incompetent comic relief or sniveling cowards). He also seemed unpopular amongst the honorable old guard like Jeong Jeong and Piandao, who probably would've been Iroh's supporters had Iroh chosen to claim his birthright.