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Added the definition of “basic triples” of octonions, and the statement that they form a torsor over $Aut(\mathbb{O}) = G_2$.
added the actual definition to octonions
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 $e_1$, $e_2$, $e_3$ 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. ]
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…
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.
added pointer also to
and will add this to various related entries, too.
have re-typed the previous example for some basic octonion algebra, now adjusted to the new labeling of generators:
$\begin{aligned} \mathrm{e}_4 \Big( \mathrm{e}_5 \big( \mathrm{e}_6 (\mathrm{e}_7 x) \big) \Big) & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \ell) x \big) \Big) \bigg) \\ & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \overline{x}) \ell \big) \Big) \bigg) \\ & = \ell \Big( (i \ell) \big( (x k) j \big) \Big) \\ & = \ell \bigg( \Big( i \big( j (k \overline{x} \big) \Big) \ell \bigg) \\ & = \big( (x k) j \big) i \\ & = \left\{ \begin{array}{ccc} \phantom{-}\, x & \text{if} & x \in \mathbb{H}\phantom{\ell} \hookrightarrow \mathbb{O} \\ - x & \text{if} & x \in \mathbb{H}\ell \hookrightarrow \mathbb{O} \end{array} \right. \end{aligned}$The definition of ’basic triple’ is wrong: if we take i, j, k in a quaternion subalgebra of the octonions it count as a basic triple according to this definition, but that’s not right. One needs to add that each one anticommutes with the product of the other two. I’ll fix this.
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