I'm not going to accept my brother's apology
Why would you not accept your brother's apology?
Because I can tell his apology wasn't sincere.
You shouldn't assume that. If someone's trying to apologize, they're trying to make things right. Why not just accept the apology?
I don't care. I'm going to make him angry and he'll try to be nice.
Forgive and forget, my friend. Don't make a bad situation worse by stoking the fire.