nLab split quaternion

Redirected from "split quaternions".

Contents

Definition

The split-quaternions are an algebra over the real numbers. Every split quaternion $q$ may be represented as

q = a_0 + a_1 i + a_2 j + a_3k

where the basis elements satisfy the following products:

$\times$	i	j	k
i	-1	k	-j
j	-k	+1	-i
k	j	i	+1

and conjugation $t^* = a_0 - a_1 i - a_2 j - a_3k$ .

These are closely related to the quaternions, as the generators satisfy similarly-looking relations. They are obtained from the split-complex numbers through the generalization of the Cayley-Dickson construction.