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.

Facts

• Bruck-Kleinfeld theorem on alternative division rings tells us that any alternative division ring that is not associative must be a Cayley-Dickson algebra.

Particular cases

• Quaternion algebra over a field is a four-dimensional algebra obtained by applying the Cayley-Dickson construction twice to a field (both applications treat the original field as the ground ring).
• Cayley octonion algebra over a field is an eight-dimensional algebra obtained by applying the Cayley-Dickson construction thrice to a field (all three applications treat the original field as the ground ring).

The case of reals

$R$	$A$	Norm function	Involution	$\lambda$	What we get for $KD(A,\lambda )$	How nice is it?
$\mathbb{R}$ -- field of real numbers	$\mathbb{R}$ -- field of real numbers	$x\mapsto x^2$	identity map	-1	$\mathbb{C}$ -- field of complex numbers	field
$\mathbb{R}$ -- field of real numbers	$\mathbb{C}$ -- field of complex numbers	$x\mapsto |x{|}^2$	$x\mapsto \overline{x}$ -- complex conjugation	-1	$\mathbb{H}$ -- quaternions	(associative) division ring
$\mathbb{R}$ -- field of real numbers	$\mathbb{H}$ -- quaternions	$x\mapsto |x{|}^2$	$x\mapsto \overline{x}$ -- conjugation in the quaternions, sending $i,j,k$ to their negatives, while fixing reals	-1	$\mathbb{O}$ -- octonions	alternative division ring
$\mathbb{R}$ -- field of real numbers	$\mathbb{O}$ -- octonions	$x\mapsto |x{|}^2$	$x\mapsto \overline{x}$ -- conjugation in the quaternions, sending all the imaginary square roots of -1 to their negatives, while fixing reals	-1	$\mathbb{S}$ -- sedenions	not an alternative division ring