Real algebras of dimension two

Question

Today I learned that there are only three possible ways to generalize $\Bbb{R}$ into algebra of dimension $2$ and all of them are given by $\{x+uy: x,y\in\Bbb{R},\text{and}\,\, u^2\in\{0,1-1\}\}.$

How can I prove this fact?
Is there any reference to know more about them?

Answer 1

We can assume $\mathbb{R}\subseteq A$, which simplifies notation.

If $t\in A\setminus\mathbb{R}$, then $\{1,t\}$ is a basis for $A$ as vector space.

In particular $t^2=a+bt$, for some $a,b\in\mathbb{R}$.

Completing the square we have $$ \left(t-\frac{b}{2}\right)^{\!2}=a+\frac{b^2}{4} $$ There are three cases:

1. $a+b^2/4>0$
2. $a+b^2/4=0$
3. $a+b^2/4<0$

Can you finish?