Let’s look a little at Alkoth. The city is the center of the Shargash cult, with roughly some 4000 cultists. It is a Lunar city and has 5000 Seven Mothers cultists – most here are going to be dedicated to an individual Seven Mother (Deezola, Yanafal Tarnils, and Irrippi Ontor probably the biggest cults). It is also one of the centers of the Yelm cult with 2000 members. There’s another 9000 adults in Alkoth – so we have lots of Dendara, Oslira, Biselenslib, Lodril and Lowfire cultists as well. I expect there might even be 150-200 Red Goddess initiates.
The ruling elite in Alkoth are going to be Lunars. Some are Yelm cultists, some are Shargash cultists – but all of the ruling elite accept the Red Emperor. In fact, this is where much of the conflict within the elite comes from: competition for the attention and favour (or disfavour) of the Red Emperor and his court (in reality it is mainly his court).
Free, family Alkorneval is an ancient family with members holding senior positions within Yelm and Shargash cult (particularly the Alkor subculture), as well several Red Goddess initiates, officers in the Yanafal Tarnils cult, Irrippi Ontor scribes, etc. They have been leaders in the city since the liberation of the city in the Fourth Wane. However, they are largely of regional ties and importance, with few contacts outside of Henjarl.
Alkorneval are locked in conflict with the Full Moon Blesses Us family, originally from the Hungry Plateau. They were installed in several key city positions about a dozen years ago, due to the intrigues of the Mother of Strength and the Tharkalists. They revere the Seven Mothers (in particular Yanafal Tarnils and Deezola), but are headed by a Jakaleel-Red Goddess shaman. The Full Moon Blesses Us family have relatively few local connections (although they were given ownership of the Raus domains in Kostaddi after a rebellion about a decade ago), but they have very good ties to Glamour and Two Tower.
These families are engaged in a low-level Dart War, with spies and assassins trying to embarrass the other family in the eyes of the Imperial Court.
So both families likely motivate themselves with their Loyalty to the Red Emperor – heck they want to show how loyal they are and how disloyal their rivals are. These conflicts are over power, status, and personal connections, and not really ideological as we would understand them.
Now if we want to throw another family into the mix, we might have some powerful administrator appointed by the Court, perhaps even a son or daughter of the Red Emperor. He has a lot of those, and this is from a previous Mask, so no particular personal connection with the current Mask. But it is a child of Moonson, and thus a grandchild of the Red Goddess (and theoretically a sibling of Jar-eel and others) so is owed respect and fealty (especially from the Yelm cult). But of course, the reality is that they need to build their own power base and has decided to start here in Alkoth.
So now you have a three-way conflict. Given the size of Alkoth, there are likely far more than this, but at least we can see the contours of elite conflict.