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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 23rd 2018

made explicit the alternative form of the definition (here) in terms of adjoining a generator and imposing relations

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 26th 2019

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 18th 2020

fixed the second relation in the definition of the CD-double via generators-and relations

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 18th 2020

added pointer to the original article:

• Leonard Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem, Annals of Mathematics Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 18th 2020

For the record and for my peace of mind, I have spelled out in full detail the computation which shows the equivalence between the component formula

$(a,b) (c, d) \;\coloneqq\; (a c - d \overline{b}, \overline{a} d + c b)$

and the relations

$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} (a \ell) b = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell) = - \overline{a b}$

As it goes, this showed that the previous version of the entry was wrong: There are two versions of the component formula, depending on whether one identifies

$(a,b) \leftrightarrow a + \ell b$

or

$(a,b) \leftrightarrow a + b \ell$

I have now changed everything to the first version. For completeness one should eventually add at least a remark about the second version.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 19th 2020

Who is the first to state the minimal set of relation

$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} (a \ell) b = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell) = - \overline{a b}$

?

Around (6) of Dickson 1919 the idea of generators $i, j, k , \ell$ appears, but not this minimal choice of set of relations.

Is this original to Baez 02, where it appears in the second half of section 2.2?