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You didn't add anything; you misunderstood the opening paragraph.
To be more constructive (in the non-mathematical sense): I don't think that this should be split up into a bunch of different contexts. It's a simple idea: imaginary = not real (for a certain sense of ‘real’), and it's the same idea for complex numbers in classical mathematics as for other hypercomplex numbers (aka elements of algebras over the real numbers) and as in constructive mathematics (although one has to be careful about how to refine the idea). So the clarification in constructive mathematics should be a parenthetical comment in the midst of the general topic, not a separate topic; and what you wrote about other *-algebras is (except for the list of examples, which is valuable) fully encapsulated in the sentence ‘This all generalizes to other kinds of hypercomplex numbers.’ (a generalization which is by no means limited to constructive mathematics). Separating these single sentences into separate sections produces a disjointed article, not useful organization. That said, some of your reorganization is a good idea; (after all, if you misunderstood the opening paragraph, then so could others).
I would not refer to elements of arbitrary *-algebras as hypercomplex numbers. So I’d say Urs is correct here.
On second thought, your sentence ‘an element of a star algebra is purely imaginary if ’ does say something not included in my sentence ‘This all generalizes to other kinds of hypercomplex numbers’, but something that I think is false. We should talk about the generalization to general *-algebras, particularly those whose elements are not generally thought of as kinds of numbers, but the term for when is not ‘purely imaginary’ but rather ‘anti-Hermitian’ (or ‘skew-Hermitian’).
@Dmitri #4:
Not arbitrary *-algebras, but arbitrary algebras. That said, one doesn't always think of these as numbers (so it's not truly arbitrary), but if they're not hypercomplex numbers, then they're not imaginary either. Or do you think that I'm wrong in #5, and you'd call, say, an anti-symmetric matrix with real entries ‘imaginary’?
I'm more concerned with the organization, however; figuring out how broadly the term is used is another matter. I do agree that, for discussing true generalizations such as anti-Hermitean matrices, we should have a separate section. So if that is what Urs is doing, then I misunderstood him and I agree with separating out that section. (But it needs some rewriting.)
Re #6: We should probably mention that in *-algebras these are known as skew-adjoint or anti-hermitian elements, not purely imaginary.
Let’s not be silly. I have adjusted it. Took me 1% the number of keystrokes as in #2-#7.
We discuss it to come to an agreement; these changes don't address my original concern much. To be sure, sometimes the best way to discuss it is to make an edit and see what people think; I hope to do that later today.
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