Posted with input from meta for improvement. I usually read, e.g. "Gaussian integers" and "Riemannian metrics", and occasionally "euclidean" or "cartesian" or even "lorentzian space", but the latter examples are relatively uncommon. I have also seen, e.g. "artinian" (and it has been commented that "algorithm" might be interpreted in a similar context, though the "al-" prefix seems to me to militate against it). I can't imagine that the relative standing of the individual has much effect, because, hey: Gaussian. Perhaps algebraists have collectively decided that spaces with some structure are fair game for the lower-case treatment?

Scott Carnahan pointed out that some people apparently subscribe to consistent conventions, e.g. "don't capitalize when the name is modified to form an adjective. These seem somewhat inconsistent with current practice, though." As Harry Gindi put it: "I think that if you can say X is (term) without the noun following it, it is generally left uncapitalized. So, notice, for instance, 'the ring X is noetherian', 'the ring X is artinian', 'the group X is abelian', 'the square F is cartesian', 'the square F is cocartesian', etc."

So, to take Deane Yang's formulation (note already the discrepancies in capitalization!): "Is there really a rational explanation why it's 'abelian', 'noetherian', and 'artinian' but 'Euclidean', 'Riemannian', and 'Lorentzian'?"