Definition 1

A Cayley algebra over A is an ordered pair (E, s), where E is an algebra over A with a unit element e and s is an antiautomorphism of E such that

for all $u \in E$.

$s$ is called the conjugation of the Cayley algebra (E, s) and $s(u)$ the conjugate of $u$. The condition $u + s(u) \in Ae$ implies that $u$ and $s(u)$ are permutable. We write

(9)

(10)

and these elements of the subalgebra $Ae$ are called respectively the Cayley trace and norm of $u$.

The ordered pair consisting of a quadratic algebra E and its conjugation s (which is an antiautomorphism since E is commutative) (no. 3) is a Cayley algebra.

Let (E, s) be a Cayley algebra; as $s(e) = e,\ s(u + s(u)) = u + s(u)$, in other words $s(u) + s^2(u) = u + s(u)$ or also

so that $s^2$ is the identity mapping of E. It follows that

Finally, the relation $(u - u)(u - s(u)) = 0$ gives

for all $u \in E$.