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 ai are Real and the bases ei 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 (eiej)ek and ei(ejek), 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?