The Emperor and House Hawkwood
At the end of the Emperor Wars were all and done - after Alexius crowned himself, after Jericho, and when the period of consolidation had really begun - the Emperor let it quietly be known that he intended on being an Emperor for all people, not just the nobles of the House of his birth.
Since then, he has been remarkably even-handed in distributing the spoils of war. While House Hawkwood has recieved a few choice positions - the Lord High Admiral of the Imperial Navy, for instance - many of the key posts in Alexius' empire have been filled by members of other Houses, including House Decados and the Hazat, the losers of the Wars. In fact, by some measures House Hawkwood has been one of the Houses to benefit the least from the Emperor Wars, despite taking some of the heaviest losses on the side of the victorious alliance.
The reason for this is obvious. While the string of military victories orchestrated by Alexius and his brother Alvarex were important to his success, it was the Emperor's prowess as a diplomat that sealed his throne and ended the Emperor Wars. He is aware that he rules not a unified Empire but a coalition of often fractious noble houses, guilds, and sects, and the only way for him to find a place in this mess unopposed by any of the power blocs already present is to rise above them - to occupy a level on the political hierarchy superior to any Prince or archbishop. To do so, he has to transcend his former position as Prince of House Hawkwood so that his victory increases no one faction's share of the power at the expense of another.
At the same time, he cannot risk alienating House Hawkwood. With six elector rods, the Hawkwood have more rods than any other House besides the Decados (with whom they are tied) and are second only to the Church in the number of rods they control - and the Patriarch only determines how a fraction of the Church's rods are cast. Given the strong emphasis on family placed by the nobility in general and the traditionalist House Hawkwood in particular, the Emperor has to walk a very fine line between being too partisan for House Hawkwood - and alienating the other Houses, collapsing the delicate coalition of peace - and turning his back entirely on his family, which would be as equally destructive to his political future.
This leaves him uniquely vulnerable to moves made against him by his House of birth, as his reaction to them must be carefully nuanced: if he reacts as Emperor to a vassal, then he turns his back on his family once and for all, but if he reacts as Prince disciplining his own then he proves his partisanship to the rest of the Known Worlds.
This difficulty is aggravated by a growing opposition to the Emperor from within his own House. This anti-Alexius faction is motivated by a frustration with the Emperor for not returning some of the spoils of victory to his own kin, and the cabal of nobles who lead it claim as their idol none other than Duke Alvarex, the Emperor's own younger brother and a great military leader during the Wars. Their desire for spoils is also not just petty: it stems from a very real disagreement with the pro-Imperial faction about the course of future events. Alexius' plan is rooted in a gamble: if he can cement his position as an Emperor who is above the internecine politics of the House, he can carve out a place for the Imperial throne that will last, bringing with it peace and a hope that humanity can struggle out of a thousand years of the New Dark Ages. There are very good reasons to believe, however, that he will fail.
Alexius's is the third attempt during the New Dark Ages to unify the Known Worlds, and all those before him - Vladimir Alecto, the first Emperor, and Halvor Li Halan, who declared himself Theocrat - have failed. Additionally, the larger scope of human history seems to indicate that such unifying endeavors can never succeed; the First Republic was destroyed by corporate infighting in the beginning days of the Diaspora, and the Second Republic grew corrupt and bloated before it, too, was torn apart from within by the same noble families who developed from those First Republic zaibatsu. It is still the same nobility who rule the Known Worlds today, and the anti-Alexius Hawkwoods reason that they have not changed - and that consequently, this Imperial experiment is doomed to fail.
Alvarex and his company - much like the hawkish parties in every House - would argue that there is only one war to end the chaos of human history, and that is not the current Empire of compromise and coalition but one of conquest, where one House sets it up as unquestioned ruler by might of arms. Towards this end, they would advocate that holding the Imperial throne by election is merely a step on the road towards true rulership, and that is an opportunity that must be used to fortify the House as best as it can. After all, the other Houses will soon enough restart their bids for power, and when that happens House Hawkwood must be prepared. To that end, the anti-Alexius Hawkwoods argue that by denying his house its just rewards Alexius is not just trying to rise above the fray but is actively obstructing the future of House Hawkwood, as at the end of the larger struggle for power there will be one House that is dominant and four others that are extinguished and shattered remnants of what they once were - and, they argue, it would be an ignoble end indeed if House Hawkwood came to pass finally into the night as a gelded housecat rather than a lion whose final roar echoed from the burnt-out cores that once were stars.