Fundamental representations and algebraic properties of biquaternions or complexified quaternions

TL;DR

This paper explores the fundamental algebraic properties and various representations of biquaternions, introducing new forms and clarifying operations like conjugation, semi-norms, and products for complexified quaternions.

Contribution

It provides a comprehensive and consistent framework for representing and manipulating biquaternions, including some novel representations and definitions.

Findings

Multiple representations of biquaternions are introduced.

Consistent notation enhances understanding of algebraic operations.

New definitions of semi-norms and polar forms are provided.

Abstract

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically.