The Origin of Storms (Lotus Kingdoms #3) by Elizabeth Bear (Tor Books June 2022)    

The Lotus Kingdoms are at war, with four claimants to the sorcerous throne of the Alchemical Emperor fielding three armies between them. Alliances are made, and broken, many times over—but in the end, only one can sit on the throne. And that one must have not only the power, but the rightful claim.