Split complex numbers C_s

The split complex algebra is $\mathbb C_s=\mathbb R[e]/(e^2-1)$ : its elements are $a+be$ , with $a,b\in\mathbb R$ , and multiplication is

(a+be)(c+de)=(ac+bd)+(ad+bc)e.

Conjugation sends $a+be$ to $a-be$ , and the multiplicative quadratic form is $N(a+be)=a^2-b^2$ .

Splitting and zero divisors

The map $a+be\mapsto(a+b,a-b)$ is a unital real-algebra isomorphism to $\mathbb R\times\mathbb R$ . Its inverse sends $(x,y)$ to $(x+y)/2+(x-y)e/2$ . The nonzero factors $1+e$ and $1-e$ multiply to zero. Thus “norm” here means an isotropic quadratic form of signature $(1,1)$ , not a positive norm and not a normed division algebra.

One-dimensional boundary

The one-dimensional composition algebra is just $\mathbb R$ , with norm $a^2$ ; there is no distinct real one-dimensional isotropic split form. In higher dimensions the split quaternion and split octonion algebras supply the other split Hurwitz algebras.

References

Alberto Elduque, Composition algebras, Section 2.1, equation (7); Theorem 2.11 and Corollary 2.12.