The Cayley-Dickson construction (a.k.a. Dickson construction or Dickson doubling) constructs a new *-algebra $A'$ out of a given *-algebra $A$. As a vector space $A' = A \oplus A$, and the multiplication and *-involution on $A'$ are customarily defined in a somewhat ad hoc manner (Wikipedia article, nlab article, Baez "The octonions" or also in the book of Schafer on Non-associative algebras).
Is there a conceptual definition of $A'$ that does not involve any elementary fiddling around (e.g. in order to prove statements such as: $A'$ is associative iff $A$ is commutative and associative)?
