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.