Cayley-Dickson construction

Definition

Suppose $R$ is a nonzero commutative unital ring and $A$ is a normed algebra over $R$, i.e., a unital non-associative (in the sense of not necessarily associative) algebra over $R$ with a norm function to $R$ (a nondegenerate quadratic form that is a multiplicative homomorphism).

Suppose there is an involution ${*}^{}$ for $A$ over $R$. This means that $(ab{)}^{*}=b^{*}{a}^{*}$ and $(a^{*}{)}^{*}=a$ for all $a,b\in A$, ${*}^{}$ is $R$-linear, and it preserves the norm function.

Suppose $\lambda$ is an invertible element of $R$. Then, we can define a new algebra $KD(A,\lambda )$ over $A$ as follows:

• As a $R$-module, $KD(A,\lambda )=A\oplus A$.
• The multiplication is defined by $(a\oplus b)(c\oplus d)=(ac+\lambda d^{*}b)\oplus (da+b{c}^{*})$
• The extended involution is defined as $(a\oplus b{)}^{*}=a^{*}\oplus (-b)$
• The extended norm is defined as $N(a\oplus b)=(a\oplus b)(a\oplus b{)}^{*}$.

This construction is termed the Cayley-Dickson construction and algebras constructed in this way are termed Cayley-Dickson algebras.