Now that you have won Boeotia for Sparta, you should return to Myrrine in front of the throne room of the capital. Speak with her to confront the Kings of Sparta, and to accuse one of them as being a cultist of Kosmos. If you were able to persuade Lagos to leave the cult, rather than kill him, you will have the proof you need to assert that Pausanias is indeed the cultist King. If not, there is no need to worry; you can still accuse Pausanias, but rather than have him banished where you can fight him on the road, the Ephors will banish you, and you must infiltrate the city and assassinate him.
In the event that Pausanias is banished, he will be surrounded by guards on the road. Use special bow skills such as Predator Shot to assassinate him from afar. If you are banished, Pausanias will be relatively unguarded within the city of Sparta. You can use Ikaros to identify his precise location, wait for him to enter a less populated area, and then attack him by surprise. Be careful not to harm any citizens in the process, as this will draw more guards to your location. Once Pausanias is dead, return to Archidamos with the evidence that you collected from the corpse of Pausanias.
Note: When you bring the proof to Archidamos, you will be rewarded with lands and citizenship returned to you. After a brief cut scene, Myrrine will ask you to return your sibling back home alive. If you want a reunited ending to the game, you should promise Myrrine that you will bring your sibling back, no matter the cost.
When you have decided how you will approach Kassandra, the quest will end.