Editing Complexification (section)

== Dickson doubling ==
{{Main|Cayley–Dickson construction}}
The process of complexification by moving from {{math|'''R'''}} to {{math|'''C'''}} was abstracted by twentieth-century mathematicians including [[Leonard Dickson]]. One starts with using the [[identity mapping]] {{math|1=''x''* = ''x''}} as a trivial [[involution (mathematics)|involution]] on {{math|'''R'''}}. Next two copies of '''R''' are used to form {{math|1=''z'' = (''a , b'')}} with the [[complex conjugation]] introduced as the involution {{math|1=''z''* = (''a'', −''b'')}}. Two elements {{mvar|w}} and {{mvar|z}} in the doubled set multiply by
:<math>w z = (a,b) \times (c,d) = (ac\ - \ d^*b,\ da \ + \ b c^*).</math>
Finally, the doubled set is given a '''norm''' {{math|1=''N''(''z'') = ''z* z''}}. When starting from {{math|'''R'''}} with the identity involution, the doubled set is {{math|'''C'''}} with the norm {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}}.
If one doubles {{math|'''C'''}}, and uses conjugation (''a,b'')* = (''a''*, –''b''), the construction yields [[quaternion]]s. Doubling again produces [[octonion]]s, also called Cayley numbers. It was at this point that Dickson in 1919 contributed to uncovering algebraic structure.

The process can also be initiated with {{math|'''C'''}} and the trivial involution {{math|1=''z''* = ''z''}}. The norm produced is simply {{math|''z''<sup>2</sup>}}, unlike the generation of {{math|'''C'''}} by doubling {{math|'''R'''}}. When this {{math|'''C'''}} is doubled it produces [[bicomplex number]]s, and doubling that produces [[biquaternion]]s, and doubling again results in [[bioctonion]]s. When the base algebra is associative, the algebra produced by this Cayley–Dickson construction is called a [[composition algebra]] since it can be shown that it has the property
:<math>N(p\,q) = N(p)\,N(q)\,.</math>