So lost or separated brothers in general, but what you love the most is enemies or rivals discovering they are brothers.
That's Legend of the Condor Heroes! Here's a good resource: Wuxia Society. The Condor Trilogy has several adaptations out there, so you should be in heaven knocking them off, one by one. I'm planning to watch the 2008 cdrama Legend of the Condor Heroes because I love Hu Ge, and I hear good things about the romance plot. But fans of the novel criticize the 2008 drama for diverging from the author's work. I recommend you look into some suggested watch orders for the film and drama adaptations and the classic novels by Jin Yong.
I second Handsome Siblings. It's a fast-moving, high action content drama with exactly what you're requesting.
Also second Pinocchio, where the brothers aren't enemies or rivals, but they are separated for years and when they first meet by chance, they don't recognize each other. Their inability to recognize each other also has plot ramifications.
Your request is mostly fulfilled by jdrama House of Ninjas. In it, there's a mysteriously missing eldest brother, Gaku Tawara, who resurfaces towards the end in a morally grey antagonist role. The drama's protagonist is the brother who continues to live in their ninja assassin "family" / clan, Haru Tawara. The ending of the series is a cliffhanger for a second series that would potentially feature an increased presence for Gaku Tawara. (I really wish Netflix would renew this, already!)
Chinese drama Guardian features brothers who are enemies, and the antagonist brother has a hidden identity. Saying even that much is a bit spoilery, but it's kinda easy to figure out. This is a wild, fast moving series that has uneven production values. There are two male protagonists who feature more prominently than the antagonist brother, and the series is based on a BL novel by Priest.
Not exactly what you're asking for, but I think you might like three dramas that feature brothers who interact on a scale from conflict to deadly enmity:
Tale of the Nine-Tailed's protagonist's brother is a persistent, volatile antagonist. This is a romance fantasy with a good amount of action.
It's Okay, That's Love has a severe rift between the protagonist and his brother. The brother does some antagonisty stuff, but the payoff reveal about their relationship doesn't come until the final episodes. For you, it would be worth waiting for, I think, and don't let anybody spoil it for you. This drama looks like a romcom, but it's pretty serious.
Six Flying Dragons has a few relationships that are adjacent to your request if not exactly what you're asking for. It's hardcore and excellent. You can either watch it with no spoilers, or you can look up the history of Yi Bang-won, and then you'll have the information that Korean audiences have going into it. I recommend you do look into his history - it'll make the drama even better to know what's coming. It's plenty suspenseful even knowing what the historical person did, because then you pay attention to how he got to that point.
