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

$\begin{aligned} \mathrm{e}_4 \Big( \mathrm{e}_5 \big( \mathrm{e}_6 (\mathrm{e}_7 x) \big) \Big) & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \ell) x \big) \Big) \bigg) \\ & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \overline{x}) \ell \big) \Big) \bigg) \\ & = \ell \Big( (i \ell) \big( (x k) j \big) \Big) \\ & = \ell \bigg( \Big( i \big( j (k \overline{x} \big) \Big) \ell \bigg) \\ & = \big( (x k) j \big) i \\ & = \left\{ \begin{array}{ccc} \phantom{-}\, x & \text{if} & x \in \mathbb{H}\phantom{\ell} \hookrightarrow \mathbb{O} \\ - x & \text{if} & x \in \mathbb{H}\ell \hookrightarrow \mathbb{O} \end{array} \right. \end{aligned}$ ]]>added pointer also to

- Tevian Dray, Corinne Manogue,
*The Geomety of Octonions*, World Scientific 2015 (doi:10.1142/8456)

and will add this to various related entries, too.

]]>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.

]]>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 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. ]

]]>Determined the remaining sign $\pm 1$ in that prop:

$L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} = + 1$ ]]>added statement and Clifford-theoretic proof (here) that the consecutive left product by all the seven imaginary generators acts as the identity

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

]]>made explicit the definition of real and imaginary octonions here

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

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

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

]]>added the actual definition to *octonions*

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

]]>