We knew the Cargyll twins would end up fighting to the death in House of the Dragon. That had to happen based on the story in Fire & Blood. I just don’t think we expected it to happen so quickly in the season.
The fight didn’t quite play out the way the book tells us it does. However, a reminder here is that the story is told from the accounts of three people. Who is to say that the accounts are accurate? It’s possible that this was always the way the fight went, but Team Black wanted people to believe that both twins kills each other so not just one of them would be viewed negatively as a kinslayer.
Being a kinslayer in Westeros is one of the worst things possible
If you watched Game of Thrones, you know that people who kill family are viewed negatively. Tyrion is constantly looked down on even more than normal after killing his father, while there are comments of Jaime killing his cousin now and then.
Being a kinslayer is one of the worst things that you can be. It doesn’t matter if that death was due to self-defense, which was the case for the Cargyll twins in House of the Dragon. Erryk and Arryk fought to the death with one protecting his queen and the other wanting to kill a pretender to the throne. There was no other way for the fight to end, but that doesn’t make it right to the people of Westeros.
So, the twin that survived would end up killing themselves. They couldn’t live with being a kinslayer, even though the death was a justified one.
Living with that death in House of the Dragons
There is also the fact that the Cargyll twins loved each other. They were both members of the King’s Guard. The only time that they split was when they both wanted to follow different monarchs. Erryk viewed Rhaenyra as the rightful queen while Arryk viewed Aegon as the rightful king. That didn’t stop them from loving each other.
We see that in the fight. Arryk didn’t even want to go to Dragonstone to pose as his twin. When they fought, they didn’t really want to kill each other. It was just a matter of need to protect/kill Rhaenyra.
So, the one who did the killing would have to live with the knowledge they killed their own twin. It didn’t even matter how Westeros viewed them after that. They had to live with that grief and guilt, and they couldn’t. So, the only option was to fall on their own sword.
It makes for a heartbreaking ending to the Cargyll twins in House of the Dragon.