  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.
