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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016
    • (edited Jan 20th 2016)

    Added the definition of “basic triples” of octonions, and the statement that they form a torsor over Aut(𝕆)=G 2Aut(\mathbb{O}) = G_2.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2016

    added the actual definition to octonions

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2018

    added statement and proof (here) that the product of all the seven imaginary quaternions with each other is ±1\pm 1.

    diff, v16, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2018
    • (edited Apr 23rd 2018)

    Added a statement (here) concerning projecting out \mathbb{H} from 𝕆\mathbb{O}.

    diff, v17, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    added statement and proof (here) that the octonions are alternative

    diff, v18, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    made explicit the definition of real and imaginary octonions here

    diff, v18, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    stated the Clifford action of the imaginary octonions induced by left multiplication (here)

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018
    • (edited Apr 24th 2018)

    added statement and Clifford-theoretic proof (here) that the consecutive left product by all the seven imaginary generators acts as the identity

    diff, v19, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2018

    Determined the remaining sign ±1\pm 1 in that prop:

    L e 7L e 6L e 5L e 4L e 3L e 2L e 1=+1 L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} = + 1

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2020
    • (edited Apr 17th 2020)

    I would like to replace the Fano plane diagram for the octonion multiplication (here) with one whose labels are more well-adapted to applications: The labels e 1e_1, e 2e_2, e 3e_3 should be on one line, and maybe best on the inner circle, so that one can readily identify them with the generators of a copy of the quaternions inside the octonions.

    Checking, I see that on Wikipedia they had just the same idea for re-labeling: here.

    But there they also reversed the direction of the straight inner lines. Hm, is that irrelevant up to isomorphism? Or is that a mistake?

    [ Never mind. I see it’s consistent with Cayley-Dickson. ]

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2020
    • (edited Apr 18th 2020)

    Okay, I have made an improved graphics of the “octonion multiplication table”, now well-adapted to inclusion of the quaternions, with the generators labeled according to their Dickson-double incarnation: here.

    Now including it into the entry…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    I have added pointer to the original references

    • Arthur Cayley, On certain results relating to quaternions, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 (doi:10.1080/14786444508562684)

    • 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)

    Then I adjusted the first sentences in the Idea-section for clarity,and then expanded the Idea-section to provide some minimum of perspective.

    diff, v28, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    added pointer also to

    and will add this to various related entries, too.

    diff, v28, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    have re-typed the previous example for some basic octonion algebra, now adjusted to the new labeling of generators:

    e 4(e 5(e 6(e 7x))) =((i)((j)((k)x))) =((i)((j)((kx¯)))) =((i)((xk)j)) =((i(j(kx¯))) =((xk)j)i ={x if x𝕆 x if x𝕆 \begin{aligned} \mathrm{e}_4 \Big( \mathrm{e}_5 \big( \mathrm{e}_6 (\mathrm{e}_7 x) \big) \Big) & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \ell) x \big) \Big) \bigg) \\ & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \overline{x}) \ell \big) \Big) \bigg) \\ & = \ell \Big( (i \ell) \big( (x k) j \big) \Big) \\ & = \ell \bigg( \Big( i \big( j (k \overline{x} \big) \Big) \ell \bigg) \\ & = \big( (x k) j \big) i \\ & = \left\{ \begin{array}{ccc} \phantom{-}\, x & \text{if} & x \in \mathbb{H}\phantom{\ell} \hookrightarrow \mathbb{O} \\ - x & \text{if} & x \in \mathbb{H}\ell \hookrightarrow \mathbb{O} \end{array} \right. \end{aligned}

    diff, v29, current

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