Editing Octonion (section)

===Cayley–Dickson construction===
{{Main|Cayley–Dickson construction}}
A more systematic way of defining the octonions is via the Cayley–Dickson construction. Applying the Cayley–Dickson construction to the quaternions produces the octonions, which can be expressed as <math>\mathbb{O}=\mathcal{CD}(\mathbb{H},1)</math>.<ref name="Ensembles">{{cite web|url=https://mathsci.kaist.ac.kr/~tambour/fichiers/publications/Ensembles_de_nombres.pdf|date=6 September 2011|title=Ensembles de nombre|publisher=Forum Futura-Science|access-date=11 October 2024|language=fr}}</ref>

Much as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions {{math|(''a'', ''b'')}} and {{math|(''c'', ''d'')}} is defined by
:<math> ( a, b )( c, d ) = ( a c - d^{*}b, da + bc^{*} )\ ,</math>
where {{math|''z''*}} denotes the [[Quaternion#Conjugation, the norm, and reciprocal|conjugate of the quaternion]] {{mvar|z}}. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs
:{{math|(1, 0), (''i'', 0), (''j'', 0), (''k'', 0), (0, 1), (0, ''i''), (0, ''j''), (0, ''k'')}}