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:

a₀e₀ + a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ + a₅e₅ + a₆e₆ + a₇e₇

Where the coefficients a_i are Real and the bases e_i have (something like) the following relations:

e₀ = 1 (and -1 = e₁² = e₂² = …)

e₁ = I = e₂e₃ = e₇e₆ = e₄e₅

e₂ = J = e₅e₇ = e₃e₁ = e₄e₆

e₃ = IJ = e₁e₂ = e₆e₅ = e₄e₇

e₄ = K = e₅e₁ = e₆e₂ = e₇e₃

e₅ = IK = e₇e₂ = e₁e₄ = e₃e₆

e₆ = JK =e₅e₃ = e₁e₇ = e₂e₄

e₇ = IJK = e₆e₁ = e₃e₄ = e₂e₅

In addition, if any of the products like e₂e₃ = e₁ are reversed you get the negative, so e₃e₂ = -e₁.

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

(e₁e₂)e₃ = e₃² = -1

e₁(e₂e₃) = e₁² = -1

(e₁e₂)e₄ = e₃e₄ = e₇

e₁(e₂e₄) = e₁e₆ = -e₇

(e₁e₂)e₅ = e₃e₅ = -e₆

e₁(e₂e₅) = e₁e₇ = e₆

so it is hit or miss I guess. Also note that e₇ = (IJ)K = -I(JK). And for all (e_ie_j)e_k and e_i(e_je_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

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

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