Construction of Hyper-Complex Numbers

Question

How does one construct a hyper-complex number multiplication table? For example:

Quarternions:

×   1   i   j   k
1   1   i   j   k
i   i   −1  k   −j
j   j   −k  −1  i
k   k   j   −i  −1

Octonions:

×   e0  e1  e2  e3  e4  e5  e6  e7
e0  e0  e1  e2  e3  e4  e5  e6  e7
e1  e1  −e0 e3  −e2 e5  −e4 −e7 e6
e2  e2  −e3 −e0 e1  e6  e7  −e4 −e5
e3  e3  e2  −e1 −e0 e7  −e6 e5  −e4
e4  e4  −e5 −e6 −e7 −e0 e1  e2  e3
e5  e5  e4  −e7 e6  −e1 −e0 −e3 e2
e6  e6  e7  e4  −e5 −e2 e3  −e0 −e1
e7  e7  −e6 e5  e4  −e3 −e2 e1  −e0

And so on.

I believe its called Cayley–Dickson_construction, but I couldn't find much information on how to construct these tables. (Or at least any information from a pattern point-of-view, all else is beyond me).

Answer 1

In general, the definition of multiplication on the $2^n$ ions is not unique. For example, there are $480$ different tables that can be constructed for the octinions.

I will adopt the tradition of writing $1 = e_0$.

Octinions and other hypercomplex numbers $e_0,\cdots, e_{2^n-1}$ must satisfy certain properties. For example, each $e_1,\cdots, e_{2^n-1}$ are square roots of $-e_0$ . The product of $e_i$ and $e_j$ are anticommunitive when $i \neq j$ and both the properties of index cycling and index doubling hold.

$$e_ie_j = e_k \implies e_{i+1}e_{j+1} = e_{k+1} \text{ (index cycling)}$$

$$e_ie_j = e_k \implies e_{2i}e_{2j} = e_{2k} \text{ (index doubling)}$$

For this, we must consider the indices as being mod $2^n -1$. As noted above, this does not define a unique table but these properties combined with one nontrivial product does define the entire table. For example, if you set $e_2e_3 = e_{5}$ your entire table is defined. If you set $e_2e_3 = e_{6}$ you have an entirely new table. I am not aware of any convention when it comes to this nontrivial product. Any convention is likely to differ between fields.