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Bioctonion

The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*:

${\displaystyle (p,q)(r,s)=(pr-s^{*}q,\ sp+qr^{*}).}$

The bioctonion z = (p,q) has conjugate z* = (p*, – q).

Then norm N(z) of bioctonion zisz 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. The algebra of bioctonions is an example of an octonion algebra.

For any pair of bioctonions y and z,

${\displaystyle N($yz)=N(y)N(z),}

showing that N is a quadratic form admitting composition, and hence the bioctonions form a composition algebra.

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 O_C of complex octonions ... The space ${\displaystyle H_{3}(O_{C})}$ of such matrices which are Hermitian with respect to the Cayley conjugation can be endowed with the structure of a Jordan algebra using a product that generalizes in a natural way the symmetrized product ${\displaystyle x\circ y={\tfrac {1}{2}}(xy+yx)}$ of ordinary square matrices. This algebra is known as the Albert algebraorexceptional Jordan algebra. 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 ${\displaystyle M_{2,1}(O_{C})}$ of 2x1 matrices with octonion entries.

Complex octonions have been used to describe the generations of quarks and leptons.^[2]

• J. D. Edmonds (1978) Nine-vectors, complex octonion/quaternion hypercomplex numbers, Lie groups and the ‘real’ world, Foundations of Physics 8(3-4): 303–11, doi:10.1007/BF00715215 link from PhilPapers.
• J. Koeplinger & V. Dzhunushaliev (2008) "Nonassociative decomposition of angular momentum operator using complex octonions", presentation at a meeting of the American Physical Society
• D.G. Kabe (1984) "Hypercomplex Multivariate Normal Distribution", Metrika 31(2):63−76 MR744966
• A.A. Eliovich & V.I. Sanyuk (2010) "Polynorms", Theoretical and Mathematics Physics 162(2) 135−48 MR2681963