Great explanation, small nitpick: in mathematics, if you leave a person's name uncapitalized it's actually a sign of respect. So we usually use the uncapitalized abelian to refer to groups which commute.
edit: This is done consistently, and I think abelian is one of the few examples. I've never seen it capitalized before, which is why I made this post.
I agree that "abelian" is often uncapitalized, but if that's a sign of respect, why is Gaussian (Gaussian elimination, Gaussian distribution, Gaussian process, etc) almost always capitalized? Or Hilbert space, Poisson distribution, Bernoulli trial, Cartesian coordinates, Euclidean distance?
It seems to me that Abelian goes uncapitalized because few people know about Niels Henrik Abel, and it should properly be capitalized.
Found this interesting MathOverflow question with some interesting possible answers. Not sure if it clears it up much, but the difference between English and French capitalization rules seems to me like it might have something to do with some of it.
My professor explained that when it gets used to the extent where people stop capitalizing it it's seen as a sign of respect. You won't see it capitalized anywhere really.
At the very least even with the knowledge that it refers to Abel they will still leave it in noun case.
Maybe it's just a post-rationalization of the phenomenon you're describing.
If it is a name it gets capitalized. If it is an adjective derived from the name (abelian, cartesian, euclidean, ...), it depends on the language, the author and the literature around the word.
No idea, but one difference I noticed with your examples are that the names are adjectives modifying a generic math term, but "abelian" is the whole label. (e.g. "abelian" vs. "Abelian group")
How is it a sign of respect? Are we not respecting Fourier for his transforms? Are we failing to respect Stokes, Green, Riemann, Cauchy, Ramanujan, and Pythagoras? The vast majority of "respected" mathematicians retain capital names on their theorems/equations/conjectures etc.
edit: This is done consistently, and I think abelian is one of the few examples. I've never seen it capitalized before, which is why I made this post.