Algebra from Cayley-Dickson Construction Never Real

Theorem

Let $A$ be a $*$-algebra.

Let $A'$ be constructed from $A$ using the Cayley-Dickson construction.

Then $A'$ is not a real algebra.

Proof

Let the conjugation operator on $A$ be $*$.

Aiming for a contradiction, suppose $A'$ is a real algebra whose conjugation operator is $*'$.

Then by definition:

$\forall a \in A': {a^*}' = a$

Let $a = \tuple {x, y} \in A'$.

Then by definition of the Cayley-Dickson construction:

$x, y \in A$

By definition of the conjugation operator:

${\tuple {x, y}^*}' = \tuple {x^*, -y}$

So for ${a^*}' = a$ it must be that $x = x^*$ and $y = -y$ from definition of Equality of Ordered Pairs.

That is, $y = 0$.

But $A'$ is a $*$-algebra, which is by definition a unitary division algebra.

So $\exists y \in A: y = 1 \ne 0$.

From that contradiction, it follows that $A'$ can not be a real algebra.

$\blacksquare$

Sources

• John C. Baez: The Octonions (2002): 2.2 The Cayley-Dickson Construction: Proposition $1$