Editing Octonion (section)

==Definition==
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real [[linear combination]] of the '''unit octonions''':

:<math>\bigl\{ e_0, e_1, e_2, e_3, e_4, e_5, e_6, e_7 \bigr\}\ ,</math>

where {{math|''e''<sub>0</sub>}} is the scalar or real element; it may be identified with the real number {{nobr| {{math|1}} .}} That is, every octonion {{mvar|x}} can be written in the form
:<math> x = x_0 e_0 + x_1 e_1 + x_2 e_2 + x_3 e_3 + x_4 e_4 + x_5 e_5 + x_6 e_6 + x_7 e_7\ ,</math>
with real coefficients {{mvar|x<sub>i</sub>}}.

===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'')}}