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Is there a package or Notebook for Mathematica that can enable me to do some numerical calculations with octonions? Maybe a way to plug-in the octonion multiplication table?

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    $\begingroup$ A Google search for "octonions mathematica" turns up library.wolfram.com/infocenter/MathSource/5194 as well as several demonstrations and multiplication tables. $\endgroup$ – whuber Mar 5 '13 at 22:01
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    $\begingroup$ If you try the link above, and it works for you, perhaps you can post an answer yourself with short review of the package. $\endgroup$ – Szabolcs Mar 6 '13 at 1:25
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Yesterday I posted THE comprehensive list of 480 octonions, Fano planes and multiplication tables here. You may be interested in the some of the code base for that, which you can find in this notebook.

It includes a Cayley-Dickson doubling of the built-in Mathematica quaternions to perform general octonion math. An interactive CDF is also available (but I suspect the users of this forum don't need that ;-)

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This is the foundation for some of the visualizations used in an interactive visualization of Octonions, E8 Lie groups and particle physics here. While I have several related Wolfram Demonstrations on Dynkin Diagrams and E8 Particles, I have been unable to successfully publish that first Octonion visualization pane.

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