Octonions are the fourth, the last, and the greatest of the so-called normed division algebras based on the Real numbers. When I first learned about the Octonions way back when, I didn’t like them because they weren’t associative like the Quaternions, the Complex numbers, and the Reals. But now I’m fine with that, and they may be important for new theories of physics!

Octonions have the general form:

a_{0}e_{0} + a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + a_{4}e_{4} + a_{5}e_{5} + a_{6}e_{6} + a_{7}e_{7}

Where the coefficients a_{i} are Real and the bases e_{i} have (something like) the following relations:

e_{0} = 1 (and -1 = e_{1}^{2} = e_{2}^{2} = …)

e_{1} = I = e_{2}e_{3} = e_{7}e_{6} = e_{4}e_{5}

e_{2} = J = e_{5}e_{7} = e_{3}e_{1} = e_{4}e_{6}

e_{3} = IJ = e_{1}e_{2} = e_{6}e_{5} = e_{4}e_{7}

e_{4} = K = e_{5}e_{1} = e_{6}e_{2} = e_{7}e_{3}

e_{5} = IK = e_{7}e_{2} = e_{1}e_{4} = e_{3}e_{6}

e_{6} = JK =e_{5}e_{3} = e_{1}e_{7} = e_{2}e_{4}

e_{7} = IJK = e_{6}e_{1} = e_{3}e_{4} = e_{2}e_{5}

In addition, if any of the products like e_{2}e_{3} = e_{1} are reversed you get the negative, so e_{3}e_{2} = -e_{1}.

Non-associativity is demonstrated by going through the list of triples:

(e_{1}e_{2})e_{3} = e_{3}^{2} = -1

e_{1}(e_{2}e_{3}) = e_{1}^{2} = -1

(e_{1}e_{2})e_{4} = e_{3}e_{4} = e_{7}

e_{1}(e_{2}e_{4}) = e_{1}e_{6} = -e_{7}

(e_{1}e_{2})e_{5} = e_{3}e_{5} = -e_{6}

e_{1}(e_{2}e_{5}) = e_{1}e_{7} = e_{6}

so it is hit or miss I guess. Also note that e_{7} = (IJ)K = -I(JK). And for all (e_{i}e_{j})e_{k} and e_{i}(e_{j}e_{k}), if they are not equal, is one equal to the negative of the other? And do I have to multiply them all out to find out?

Further Reading:

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

http://mathworld.wolfram.com/Octonion.html

http://homepages.wmich.edu/~drichter/octonions.htm

https://ncatlab.org/nlab/show/normed+division+algebra

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