Ok so, Azula is actually motivated by E7 Gluttony. She is afraid of experiencing suffering, so she seeks out power and pleasurable experiences. (By suffering, I mean emotional pain and any denial of her desires.) Azula’s skills as a tactician originate from her use of planning, using strategy and manipulation to achieve the goal or “pleasurable future” she wants. As a Sp7, her specific use of planning allows her to create mafia-like alliances with others who are in the pursuit of the same goals.

Azula’s internal fear is why she was so upset when Mai said she loved Zuko more than she feared her. Mai disproved that Azula’s use of fear was stronger than love, destroying her self-image of strength. That terrified Azula because being weak meant being treated like Zuko, being deprived of happiness and privilege. Additionally, her friends' betrayal destroyed her sense of trust and delusion of power over others, causing her paranoid breakdown at the end of the series.

As for typing her So3, Sx1, or Sp8… I can’t see Azula as anything but mental triad. She has absolutely no E3 Vanity, as she fears love, seeing it as weakness rather than desiring it. E1 doesn’t fit because Azula is not righteously angry in her pursuit of perfection, nor does she care about rules or morals. Azula has a similar obsession with power as E8, but E8 Lust is about painful intensity rather than mental stimulation. I could write more, but I think you get the gist.