The term "Illuminati" is actually shorthand from "Bavarian Illuminati", as that's where the secret society was founded. https://en.m.wikipedia.org/wiki/Illuminati
Base +6, Noxmare +6, darkbiden +6, Lao Shan Long +9, Lumberjac108 +7
Well, they're only the main antagonist of the fourth Symphogear show, but they're mostly responsible for the third and a trio of their random cast offs are still villains for almost the entirety of the final one.
No, the Bavarian Illuminati is the overarching antagonist of the series. Like they're in the background for everything since S1 either providing funding or giving the antagonists access to their doomsday devices. Even in S1, Fine's actions are largely enabled through assistance of them-- at least as a result of future retcon