Definition:Real Element in Star-Algebra

Definition
Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

Let $A' = \struct {A_F, \oplus'}$ be constructed from $A$ using the Cayley-Dickson construction.

Let $a \in A$ be real.

Then $\tuple {a, 0}$ is defined as real in $A'$.