Cayley octonion algebra

Definition

Suppose $K$ is a field. Suppose $\lambda _{1},\lambda _{2},\lambda _{3}$ are (possibly equal, possibly distinct) nonzero elements of $K$. The Cayley octonion algebra over $K$ with parameters $\lambda _{1},\lambda _{2},\lambda _{3}$ is defined as:

$KD(KD(KD(K,\lambda _{1}),\lambda _{2}),\lambda _{3})$

where $KD$ is the Cayley-Dickson construction, and where:

• In all three stages, the ground ring is $K$.
• The norm map on $K$ that we begin with is the map $x\mapsto x^{2}$.
• The involution on $K$ that we begin with is the identity map.

Cayley octonion algebras are sometimes simply called Cayley algebras.

See also octonion algebra and split octonion algebra.