In the foregoing discussion, an algebra shall mean a non-associative algebra.

Let $A$ be a normed $*$ -algebra, an algebra admitting an involution $*$ , over a commutative ring $R$ with $1\neq0$ . The Cayley-Dickson 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) $N(a\oplus b):=(a\oplus b)(a\oplus b)^*$ .

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 well-defined 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 $a\oplus0$ can now be written simply as $a$ . Now, the extended norm $$N(a\oplus b)=(a\oplus b)(a^*\oplus(-b))=(aa^*-\lambda b^*b)\oplus0=N(a)-\lambda N(b)\in A,$$ 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 $N(0\oplus0)=0$ show that the extended norm $N$ on $KD(A)$ is well-defined. Thus, $KD(A)$ is a normed $*$ -algebra.

The normed $*$ -algebra $KD(A)$ , together with the invertible element $\lambda\in R$ , is called the Cayley-Dickson algebra, $KD(A,\lambda)$ , obtained from $A$ .

If $A$ has a unity 1, then so does $KD(A,\lambda)$ and its unity is $1\oplus0$ . Furthermore, write $i=0\oplus1$ , we check that, $ia=(0\oplus1)(a\oplus0)=0\oplus a^*=(a^*\oplus0)(0\oplus1)=a^*i$ . 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.
• $KD(\mathbb{O},-1)=\mathbb{S}$ , which are called the sedenions, an algebra of dimension 16 over $\mathbb{R}$ .

Remark. Starting from $\mathbb{R}$ , notice each stage of Cayley-Dickson 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!