Yeah, just got home and couldn't at work.
The Atrisian Empire is a sovereign state, who has signed a treaty with the Republic in
this thread to become a protectorate of the Republic. This means the Empire is still a completely separate self governing body, but if it was ever attacked the Republic would help defend it. In exchange Atrisia promised trade deals and that its military would fall under command of the Republic. The Atrisian military is still maintained and managed by Atrisia (so Atrisian ships are crewed by Atrisians, and do not serve on Republic ships), but they are under the command of Akio who answers to the Republic Admiral of Fleets.
In this way Atrisian military personnel like Akio are also part of the Republic faction, while others like members of the
Reki would not be. The idea is that the Atrisian Empire, while having their own identity, fundamentally use an Imperial style of government but are close allies to the Republic. Republic and Atrisian Empire ships fighting side by side. This is part of where what makes the Atrisian Empire interesting. That and the internal discord of the Imperial style government obsessed with law and the underground Reki who keep things running in their own way.
Hopes this clears things up.