There are various ways to think about the pyreen. Whether you accept the retcon that Rajaat is a pyreen is pivotal.
If Rajaat is a pyreen: the pyreen are a chastened race. The pyreen were supposed guide the other races towards a healthy, civilized existence but instead one of their number brought about the near destruction of their world.
In this scenario, Rajaat likely had pyreen collaborators. Perhaps one of these may even have been Lalali-Puy. Now the pyreen do not form factions. They avoid hubris and do not involve themselves in world changing plots, because that is where Rajaat and his followers went wrong. Instead they work on their own immediate surroundings.
If Rajaat is NOT a pyreen: In this scenario I suppose you could have factions, although the general lore of pyreen tends to work against it. Instead of naming factions, you can start with their stance towards the world. Active/passive, aggressive/pacifist, anti sorcerer monarch/work within the system.
In my view, the pyreen are loners that come together only to mate. They aren’t a conventional race. Their ways are not our ways.