CayleyDickson construction
In the foregoing discussion, an algebra^{} shall mean a nonassociative algebra.
Let $A$ be a normed $*$algebra, an algebra admitting an involution^{} (http://planetmath.org/Involution2) $*$, over a commutative ring $R$ with $1\ne 0$. The CayleyDickson construction is a way of enlarging $A$ to a new algebra, $KD(A)$, extending the $*$ as well as the norm operations in $A$, such that $A$ is a subalgebra^{} of $KD(A)$.
Define $KD(A)$ to be the module (external) direct sum^{} of $A$ with itself:
$$KD(A):=A\oplus A.$$ 
Therefore, addition in $KD(A)$ is defined by addition componentwise in each copy of $A$. Next, let $\lambda $ be a unit in $R$ and define three additional operations:

1.
(Multiplication) $(a\oplus b)(c\oplus d):=(ac+\lambda {d}^{*}b)\oplus (da+b{c}^{*})$, where $*$ is the involution on $A$,

2.
(Extended involution) ${(a\oplus b)}^{*}:={a}^{*}\oplus (b)$, and

3.
(Extended Norm) $N(a\oplus b):=(a\oplus b){(a\oplus b)}^{*}$.
One readily checks that the multiplication is bilinear^{}, since the involution $*$ (on $A$) is linear. Therefore, $KD(A)$ is an algebra.
Furthermore, since the extended involution $*$ is clearly bijective and linear, and that
$${(a\oplus b)}^{**}={({a}^{*}\oplus (b))}^{*}={a}^{**}\oplus b=a\oplus b,$$ 
this extended involution is welldefined and so $KD(A)$ is in addition a $*$algebra.
Finally, to see that $KD(A)$ is a normed $*$algebra, we identify $A$ as the first component^{} of $KD(A)$, then $A$ becomes a subalgebra of $KD(A)$ and elements of the form $a\oplus 0$ can now be written simply as $a$. Now, the extended norm
$$N(a\oplus b)=(a\oplus b)({a}^{*}\oplus (b))=(a{a}^{*}\lambda {b}^{*}b)\oplus 0=N(a)\lambda N(b)\in A,$$ 
where $N$ in the subsequent terms of the above equation array is the norm on $A$ given by $N(a)=a{a}^{*}$. The fact that the $N:KD(A)\to A$, together with the equality $N(0\oplus 0)=0$ show that the extended norm $N$ on $KD(A)$ is welldefined. Thus, $KD(A)$ is a normed $*$algebra.
The normed $*$algebra $KD(A)$, together with the invertible element $\lambda \in R$, is called the CayleyDickson algebra, $KD(A,\lambda )$, obtained from $A$.
If $A$ has a unity 1, then so does $KD(A,\lambda )$ and its unity is $1\oplus 0$. Furthermore, write $i=0\oplus 1$, we check that, $ia=(0\oplus 1)(a\oplus 0)=0\oplus {a}^{*}=({a}^{*}\oplus 0)(0\oplus 1)={a}^{*}i$. Therefore, $iA=Ai$ and we can identify the second component of $KD(A,\lambda )$ with $Ai$ and write elements of $Ai$ as $ai$ for $a\in A$.
It is not hard to see that $A(Ai)=(Ai)A\subseteq Ai$ and $(Ai)(Ai)\subseteq A$. We are now able to write
$$KD(A,\lambda )=A\oplus Ai,$$ 
where each element $x\in KD(A,\lambda )$ has a unique expression $x=a+bi$.
Properties. Let $x,y,z$ will be general elements of $KD(A,\lambda )$.

1.
${(xy)}^{*}={y}^{*}{x}^{*}$,

2.
$x+{x}^{*}\in A$,

3.
$N(xy)=N(x)N(y)$.
Examples. All examples considered below have ground ring the reals $\mathbb{R}$.

•
$KD(\mathbb{R},1)=\u2102$, the complex numbers^{}.

•
$KD(\u2102,1)=\mathbb{H}$, the quaternions.

•
$KD(\mathbb{H},1)=\mathbb{O}$, the octonions^{}.

•
$KD(\mathbb{O},1)=\mathbb{S}$, which are called the sedenions, an algebra of dimension 16 over $\mathbb{R}$.
Remarks.

1.
Starting from $\mathbb{R}$, notice each stage of CayleyDickson construction produces a new algebra that loses some intrinsic properties of the previous one: $\u2102$ is no longer orderable (or formally real); commutativity is lost in $\mathbb{H}$; associativity is gone from $\mathbb{O}$; and finally, $\mathbb{S}$ is not even a division algebra^{} anymore!

2.
More generally, given any field $k$, any algebra obtained by applying the CayleyDickson construction twice to $k$ is called a quaternion algebra over $k$, of which $\mathbb{H}$ is an example. In other words, a quaternion algebra has the form
$$KD(KD(k,{\lambda}_{1}),{\lambda}_{2}),$$ where each ${\lambda}_{i}\in {k}^{*}:=k\{0\}$. Any algebra obtained by applying the CayleyDickson construction three times to $k$ is called a Cayley algebra, of which $\mathbb{O}$ is an example. In other words, a Cayley algebra has the form
$$KD(KD(KD(k,{\lambda}_{1}),{\lambda}_{2}),{\lambda}_{3}),$$ where each ${\lambda}_{i}\in {k}^{*}$. A Cayley algebra is an octonion algebra when ${\lambda}_{1}={\lambda}_{2}={\lambda}_{3}=1$.
References
 1 Richard D. Schafer, An Introduction to Nonassociative Algebras, Dover Publications, (1995).
Title  CayleyDickson construction 
Canonical name  CayleyDicksonConstruction 
Date of creation  20130322 14:54:11 
Last modified on  20130322 14:54:11 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  25 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 17A99 
Synonym  CayleyDickson process 
Synonym  doubling process 
Synonym  octonion algebra 
Related topic  TheoremsOnSumsOfSquares 
Defines  CayleyDickson algebra 
Defines  sedenion 
Defines  quaternion algebra 
Defines  Cayley algebra 