In medieval times, two things were commonplace: trial by combat and family dysfunction. Throughout medieval history, noble families often fought one another for power and status. Easily one of the most extreme examples was the War of the Roses, which resulted in a 30 year blood feud between York and Lancaster, who were both related to one another and both coveted the throne of England. Trial by combat was often a way for people to settle disputes in a more fair and civilized fashion.
In my fantasy world, the family that rules the Gradaian Empire are a family of dragon riders. So far, they have not developed an excuse to fight one another, but since blood feuds were common in medieval times, a feud between this powerful family seems inevitable. With this in mind, a civil war between dragon riders would definitely result in an amount of death and destruction that would take generations to recover from. So to avoid such a volatile powder keg, a more civilized approach should be developed in case these dragon riders have disputes to settle.
Imagine a trial by combat between dragon riders where one dragon rider battles against another. There would be no outside help. Like a normal trial by combat, the duel is over when a combatant is either killed or yields. The only catch is that you need to have your dragon able to fight and fly until the end of the duel. If you lose your dragon, you are disqualified and you automatically lose the match. Such a battle would light up the sky like a second sun and the sound of roaring and screaming would fill the air like a swarm of damned souls.
With such a system in place, a civil war involving dragons would be avoided and family disputes can be settled fairly and evenly.