Editing Cayley–Dickson construction (section)

== General Cayley–Dickson construction ==
{{harvtxt|Albert|1942|p= 171}} gave a slight generalization, defining the product and involution on {{math|''B'' {{=}} ''A'' ⊕ ''A''}} for {{mvar|A}} an [[*-algebra|algebra with involution]] (with {{math|(''xy'')* {{=}} ''y''*''x''*}}) to be 
: <math>\begin{align} (p, q) (r, s) &= (p r - \gamma s^* q, s p + q r^*)\, \\ (p, q)^* &= (p^*, -q)\, \end{align}</math>
for {{mvar|γ}} an additive map that commutes with {{math|*}} and left and right multiplication by any element. (Over the reals all choices of {{mvar|γ}} are equivalent to −1, 0 or 1.) In this construction, {{mvar|A}} is an algebra with involution, meaning:
* {{mvar|A}} is an [[abelian group]] under {{math|+}}
* {{mvar|A}} has a product that is left and right [[distributive property|distributive]] over {{math|+}}
* {{mvar|A}} has an involution {{math|*}}, with {{math|(''x''*)* {{=}} ''x''}}, {{math|(''x'' + ''y'')* {{=}} ''x''* + ''y''*}},  {{math|(''xy'')* {{=}} ''y''*''x''*}}.
The algebra {{math|''B'' {{=}} ''A'' ⊕ ''A''}} produced by the Cayley–Dickson construction is also an algebra with involution.

{{mvar|B}} inherits properties from {{mvar|A}} unchanged as follows. 
* If {{mvar|A}} has an identity {{math|1<sub>''A''</sub>}}, then {{mvar|B}} has an identity {{math|(1<sub>''A''</sub>, 0)}}.
* If {{mvar|A}} has the property that {{math|''x'' + ''x''*}}, {{math|''xx''*}} associate and commute with all elements, then so does {{mvar|B}}. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.
 
Other properties of {{mvar|A}} only induce weaker properties of {{mvar|B}}:
* If {{mvar|A}} is commutative and has trivial involution, then {{mvar|B}} is commutative.
* If {{mvar|A}} is commutative and associative then {{mvar|B}} is associative.
* If {{mvar|A}} is associative and {{math|''x'' + ''x''*}}, {{math|''xx''*}} associate and commute with everything, then {{mvar|B}} is an [[alternative algebra]].