Maths - Cayley Table - 2D

If we assume a 2D algebra, one possible algebra that can be represented is complex numbers, but are there any other types of algebra that are valid groups?

let the 2 dimensions be e1 & e2.

In order for the tables to obey the rules of a group (identity element, one element in each row & column, etc)

e1 e2
e1 e1 e2
e2 e2 e1

The table only represents the type of each entry - not yet whether there is any sign reversal.

There are only two valid possibilities, for a valid group, e22 = +e1 equivalent to dual and e22= -e1 equivalent to complex numbers.

There are a number of questions that occur to me about this, for instance:

So if we start from the squares of the dimensions, we either have the dual

or the complex numbers.

I cant think of a way to derive the other terms from these terms?

If we include e1 * e2 = e2 as a given fact, can we derive the last entry: e2 * e1 = e2 ?

We can try this for both cases, as follows:

since: e2 = e1*e2

square both sides:

e22 = e1*e2*e1*e2

but also we can put one or more e1* in front

e22 = e1*e1*e22

therefore this gives e1*e2 = e2*e1

This is true regardless of whether e22 = +e1 or -e1 so it applies to all cases:

dual complex
e1 e2
e2 e1
e1 e2
e2 -e1

So we can derive the right identity operator from the left identity operator, but this seems to be the only thing that is connected. The other results seem to be independent of each other?

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