The Fallout games have showcased several different chapters of the Brotherhood of Steel, and each one has been different in its own way. While described as being "a little fanatical" even in the isometric Fallout days, the Amazon show's chapter seemed particularly heavy on the olde worlde religious overtones. According to a self-described "brainrot theory" suggested by OmegaSpartan256 on the , maybe there's a good reason for that. What if the West Coast 
chapter of the Brotherhood seen in the [[link]] show synthesized the beliefs of Caesar's Legion?
As they put it, [[link]] "Maximus? Titus? Thaddeus? Quintus? All names you'd hear assigned to Legion raised individuals." In the games we've met members of the Brotherhood of Steel who have names with a little mythic resonance, like Jacob, Mary, Arthur, and Tristan, but they've just as often been given more typical names, "never Latin/Roman" as OmegaSpartan256 points out.
"Also," they go on, "just like how the Legion erased identities of the tribes it would conquer, the Brotherhood erased the Legion but only in name. They retained a lot of what the Legion would be known for or at least would very likely accept as customs."
All that incense and branding would make sense as a way of appealing to members of the old-fashioned Legion [[link]] you were looking to synthesize into your Power Armor Club. But the "big kicker", as this theory has it, comes down to the flag. 
In the show, the Brotherhood has a red-and-yellow flag, a color combination we've never seen on the Brotherhood before. (They even reserve red for knights, treating it as extra-symbolic.) Who has a red-and-yellow flag in the Fallout games? Caesar's Legion, of course.
I'm not saying I believe this theory is 100% true, and neither is OmegaSpartan256, who jokes they took a bunch of "Psycho, Mentats, and Med-X before typing this up". But this is a show being executive-produced by Jonathan Nolan, and if you've seen his other show Westworld—which Fallout resembles more than a little—you know he loves a twist with a long setup.
The West Coast Brotherhood may not have absorbed the remnants of the Legion after the events of New Vegas, though it would give another reason why they hate the NCR so much. Still, I wouldn't put it past Nolan to embed some kind of twist into Fallout we haven't uncovered yet. Especially since The Gamers have been too busy obsessing over a instead of actually watching what's not only a great bit of TV, but one that's clearly in love with all of the games. Yes, including New Vegas.
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