Advances in Applied Clifford Algebras

, Volume 21, Issue 3, pp 607–636

Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions

  • Stephen J. Sangwine
  • Todd A. Ell
  • Nicolas Le Bihan

DOI: 10.1007/s00006-010-0263-3

Cite this article as:
Sangwine, S.J., Ell, T.A. & Le Bihan, N. Adv. Appl. Clifford Algebras (2011) 21: 607. doi:10.1007/s00006-010-0263-3


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 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.

Mathematics Subject Classification (2010)

Primary 11R52Secondary 15A66



Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Stephen J. Sangwine
    • 1
  • Todd A. Ell
    • 2
  • Nicolas Le Bihan
    • 3
  1. 1.School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUnited Kingdom
  2. 2.SavageUSA
  3. 3.GIPSA-Lab, Département Images et Signal, Domaine UniversitaireSaint Martin d’Hères cedexFrance