Then norm N(z) of bioctonion z is z z* = p p* + q q*, which is a complex quadratic form with eight terms.
The bioctonion algebra is sometimes introduced as simply the complexification of real octonions, but in abstract algebra it is the result of the Cayley–Dickson construction that begins with the field of complex numbers, the trivial involution, and quadratic form z2.
Guy Roos explained how bioctonions are used to present the exceptional symmetric domains:[1] The explicit description of the exceptional domains ... involves 3x3 matrices with entries in the Cayley-Graves algebra OC of complex octonions ...
It is the natural place to describe the exceptional symmetric domain of dimension 27.
The second exceptional symmetric domain (of complex dimension 16) lives in the space