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CayleyDickson construction
In the foregoing discussion, an algebra shall mean a nonassociative algebra.
Let $A$ be a normed $*$algebra, an algebra admitting an involution $*$, over a commutative ring $R$ with $1\neq 0$. 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) $(a\oplus b)(c\oplus d):=(ac+\lambda d^{*}b)\oplus(da+bc^{*})$, 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
$N(a\oplus b)=(a\oplus b)(a^{*}\oplus(b))=(aa^{*}\lambda b^{*}b)\oplus 0=N(a)% \lambda N(b)\in A,$ 
where $N$ in the subsequent terms of the above equation array is the norm on $A$ given by $N(a)=aa^{*}$. The fact that the $N\colon KD(A)\to A$, 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 $\mathbb{R}$.

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

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

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