Definition:Real Element in Star-Algebra

Definition
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.

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

Let $a \in A$ be real.

Then $\left({a, 0}\right)$ is defined as real in $A'$.