CayleyDickson Construction:
This is a process by which a 2^{n}dimensional
hypercomplex number is constructed from a pair of 2^{n1}dimensional
hypercomplex numbers, where n is a positive integer. This is accomplished
by defining the multiplication rule for the two 2^{n}dimensional
hypercomplex numbers in terms of the four 2^{n1}dimensional
hypercomplex numbers.
Let X and
Y be two 2^{n}dimensional hypercomplex numbers.
Write each X and Y as an ordered pair of 2^{n1}dimensional
hypercomplex numbers, A, B, C, and D,
X := ( A ; B ) ,
Y := ( C ; D ) .
Then the CayleyDickson process defines the multiplication
of X and Y by
X Y := ( A ; B ) ( C ;
D ) := ( AC + mD*B ; BC* + DA )
,
where D* is a conjugate of D and m
is
a field parameter.
2dimensional complex numbers (n=1),
4dimensional quaternions (n=2), 8dimensional octonions (n=3), 16dimensional
sedenions (n=4), 32dimensional hypercomplex numbers (n=5), etc., can all
be constructed from real numbers by the iterations of this process.
At each iteration some new basal elements , i_{k },
are introduced with the property i_{k}^{2}
= 1.
In the following we will choose the field parameter m
= 1 and construct

complex numbers from real numbers,

quaternions from complex numbers,

octonions from quaternions.
It is too easy to run out of letters to use so some letters
in one section may represent different things in another.
[I] Complex Numbers (n=1) :
Consider two complex numbers,
Z = a + b i_{1} , and W = c +
d i_{1} ,
where a , b, c, d e
R
,
and ( i_{1})^{2}
= 1.
(The usual notation for the imaginary number is i instead
of i_{1 }) . Then
Z W = ( ac  bd ) + ( ad + bc ) i_{1}
Compare this with CayleyDickson multiplication rule
for Z = ( a ; b ), and W = ( c ; d ),
with_{ }m = 1
,
Z W = ( a ; b ) ( c ; d ) = ( ac 
bd ; ad + bc ) , d* = d, since d e R
,
which clearly shows that ( ac  bd ; 0 ) and ( 0 ; ad
+ bc ) correspond to ( ac  bd ) and (ad + bc) i_{1 },
respectively.
Note : If i_{2} is another
imaginary number with the property ( i_{2 })^{2}=
1, but different from i_{1} such that
i_{2}
i_{1} =  i_{1}
i_{2} , then
i_{2}
Z = i_{2} ( a + b i_{1
})
= a i_{2}  b i_{1}
i_{2} = Z* i_{2}
.
[II] Quaternions (n=2) :
In order to construct a quaternion
we need to introduce another imaginary unit, i_{2}
, with the property i_{2}^{2 }=
1, but different from i_{1} . Write
two quaternions, P, and Q, each in terms of two complex numbers,
P := A + B i_{2}
, Q := C + D i_{2}
,
where A, B, C, D e C
, and so A, B, C, D commute. Then
PQ = ( A + B i_{2 })(
C + D i_{2 }) = AC + B i_{2}
C + AD i_{2} + B i_{2}
D i_{2}
= AC + BC* i_{2} + DA i_{2}
+ BD* i_{2} i_{2}
= ( AC  D*B ) + ( BC* + DA ) i_{2}
.
Compare this with the CayleyDickson multiplication of
two quaternions defined as ordered pairs of complex numbers, ( A ; B )
and ( C ; D ), with m = 1,
PQ := ( A ; B )( C ; D ) = ( AC 
D*B ; BC* + DA )
Note that in the expression for P := A + B i_{2}
, there is a term consisting of i_{1} i_{2}
.
This represents a new imaginary number, a basal element
of quaternions, i_{3} := i_{1}
i_{2} .
[III] Octonions (n=3) :
Let P, Q, R,
S be four quaternions. Define two octonions, M and N, as the ordered pairs,
M := ( P ; Q ),
N := ( R ; S ) . Then the multiplication of M and N is given
by
MN := ( PR + m
S*Q
; QR* + SP ).
The new basal elements introduced in the process are
i_{4 }, i_{5 },
i_{6 }, and i_{7}
. 