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non associative algebras – Conceptual acceptation of the Dickson development Answer

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non associative algebras – Conceptual acceptation of the Dickson development

The Cayley-Dickson development (a.okay.a. Dickson development or Dickson doubling) constructs a fresh *-algebra $A’$ out of a given *-algebra $A$. As a vector house $A’ = A oplus A$, and the multiplication and *-involution on $A’$ are usually outlined in a considerably advert hoc method (Wikipedia article, nlab article, Baez “The octonions” or too within the bespeak of Schafer on Non-associative algebras).

Is there a conceptual definition of $A’$ that doesn’t contain any elementary fiddling round (e.g. in organize to show statements corresponding to: $A’$ is associative iff $A$ is commutative and associative)?

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