Determination of the biquaternion divisors of zero, including the idempotents and nilpotents

TL;DR

This paper provides explicit formulas and modern notation for identifying all divisors of zero, including idempotents and nilpotents, in the biquaternion algebra, clarifying their properties and relationships.

Contribution

It introduces simple formulae for biquaternion divisors of zero, unifies their properties using modern notation, and clarifies the conditions for their existence and composition.

Findings

Explicit formulas for idempotents, nilpotents, and divisors of zero

Conditions for a biquaternion to be a divisor of zero

Numerical examples illustrating the concepts

Abstract

The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.