>>11169935>I understand how the squares of i j and k equal negative one but not how ijk is equal as wellIt's defined that way. you could similarly just define quaternions with ij=k, and end up with the same structure.
>why a triple doesn’t work and why hexatonions are not a thing.Something, something Cayley-Dickson construction, something, something Frobenius theorem... Basically, you can have any dimensional algebra you want, but if you want an algebra over the reals, then you can only get a real finite dimension division algebra with R, C, H, and O. You can make a three dimensional real algebra, but there are going to nonzero elements which multiply to give zero.
You can also make division algebras with whatever dimension you want, but it will not be over the reals, it would be e.g. over the rationals with a square root of some number or something. In this case, you miss completeness, and an embedding of R inside the algebra.
Basically, you can invent whatever wacky algebra you want pretty easily, but only particular algebras are going to have the characteristics you want to make them useful.
