If the king is gone, then the princes would fight for the throne too; just like the king who killed his two brothers. Hence, the history will repeat itself.
Of course, by law, the first son will ascend to the throne, but there was no law that the other sons should not or could not fight for the said throne.
In that brutal lineage, dropping out of succession is dropping your royalty.
No money, no house, no land— just a plain old commoner who cannot be a noble unless they join the knighthood. Plus, judging the way the princes and princesses are raised, I don't think they have a good bond to even join hands, or even care much for their mother. (It's normal to have a wet maid that breastfeed the young noble/royal bc it was deemed unladylike and commoner act o breast feed your baby.)
So I guess it's clear why they didn't join hands and fight against the king.
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I have no hostility towards you, jus trying to explain why I know of and why I think that way won't be suitable for their situation.
Why not kill the king? Like if I was one of the princes, after seeing him murder my mother, I would just off him. Because why play his stupid succession game? If it was me, I would've just had a secret meeting with the rest of the Royal children and been like I'll drop out of the succession race but let's kill the king. Idgaf who becomes king but that man's gotta go.
Maybe there's a reason why they didn't kill him so I'll withhold my judgement.