CAYLEY-DICKSON ALGEBRAS
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Cayley-Dickson Construction:

This is a process by which a 2n-dimensional hypercomplex number is constructed from a pair of  2n-1-dimensional hypercomplex numbers, where n is a positive integer. This is accomplished by defining the multiplication rule for the two 2n-dimensional hypercomplex numbers in terms of the four  2n-1-dimensional hypercomplex numbers.

    Let X and Y be two 2n-dimensional hypercomplex numbers.   Write each X and Y as an ordered pair of 2n-1-dimensional hypercomplex numbers, A, B, C, and D,
    X  :=  ( A ; B ) ,     Y :=  ( C ; D ) .
Then the Cayley-Dickson 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.

    2-dimensional complex numbers (n=1), 4-dimensional quaternions (n=2), 8-dimensional octonions (n=3), 16-dimensional sedenions (n=4), 32-dimensional 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 , ik , are introduced with the property ik2 = -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 i1  , and  W = c + d i1 ,
    where a , b, c, d e R , and  ( i1)2 = -1.
(The usual notation for the imaginary number is i instead of i1 ) . Then
    Z W = ( ac - bd ) + ( ad + bc ) i1
Compare this with Cayley-Dickson 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) i1 , respectively.
Note : If i2 is another imaginary number with the property ( i2 )2= -1, but different from i1 such that
    i2 i1  =  -  i1 i2  , then
    i2 Z  =  i2  ( a + b i1 )  =  a i2  -  b  i1 i2  =  Z* i2  .

[II]    Quaternions (n=2) :
    In order to construct a quaternion we need to introduce another imaginary unit, i2 , with the property  i22  = -1, but different from  i1 . Write two quaternions, P, and Q, each in terms of two complex numbers,
    P := A + B i2 ,    Q := C + D i2  ,
where A, B, C, D e  C , and so A, B, C, D commute. Then
    PQ = ( A + B i2 )( C + D i2 ) = AC + B i2 C + AD i2  + B i2 D i2
          = AC +  BC* i2  +  DA i2   + BD* i2 i2    =  ( AC - D*B ) + ( BC* + DA ) i2 .
Compare this with the Cayley-Dickson 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 i2 , there is a term consisting of i1 i2  .
This represents a new imaginary number, a basal element of quaternions,  i3 :=  i1 i2 .

[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 i4 , i5 , i6 , and i7 .
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