\begin{definition}[Definition:Real Element in Star-Algebra]
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 $A'$.
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\end{definition}