There doesn't seem to be a lot of agreement on Brummell/brummel . Brummel seems to be common, but New England Ropes uses Brummell. I would think that the origin of the name would matter.
I've always assumed that it comes from Beau Brummell, since the eye splice looks so much like a necktie, in which case, of course, it would be with the double L. In fact, when I first saw the name in one of Brion's books, I figured that he had named it. It just seemed to have his flair for the language. (I have since learned that he didn't, but I still think it looks like his work.
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Best wishes,
Mike