(1) Quaternionic representation:
An octonion, A, is written as an ordered pair of two 4-dimensional quaternions, Q₁ and Q₂ as
        A := ( Q₁; Q₂ )  .
For the algebra of octonions, O , to satisfy the basic properties outlined in Introduction one needs to define the multiplication rules of two octonions,  A and B.  Writing B in terms of two quaternions, Q₃ and Q₄, we define the multiplication of A and B, via Cayley-Dickson process by
        AB := ( Q₁; Q₂ )( Q₃; Q₄ )
                := ( Q₁Q₃ + bQ₄*Q₂; Q₂Q₃* + Q₄Q₁ ) ,
where Q* is a conjugate of Q, and  b is a field parameter.

(2) Coordinate representation:
    An octonion, A, may be written componentwise as
        A := _m=0S⁷  a_m i_m  =  a₀ i₀  +  _j=1S⁷  a_j i_j ,
where im are the basal elements of O. The basal element, i₀, is chosen as  the basis of reals, R.  For the algebra of octonions to satisfy the basic properties outlined in Introduction one needs to define the multiplication rules for the basis. There are many choices for this task but we will describe just two of them below.
     (I) via congruence modulo 7:
            Define     i_j i_j = -1 ;    i_j i_j+1 :=  i_j+3 :=  i_j-4 ,        (mod 7)
               with         i_2k i_2m = i_2n,    whenever i_k i_m = i_n .
Then the multiplication rule for the seven basal elements, i_j  ( j = 1,...7) is summed up in the following seven cycles:
        (1,2,4) ,  (1,3,7) ,  (1,5,6) ,  (2,3,5) ,  (2,6,7) ,  (3,4,6) ,  (4,5,7) .
The cycle (j,k,m) is to mean

        i_ji_k= i_m ,   i_k i_m = i_j ,   i_k i_j = - i_m  .

    (II) via Cayley-Dickson process from the quaternion basis:
Write octonion basal element as ordered pair of quaternion basal elements
    i_m  := ( i_m ; 0 ) ,      and       i_m+4  := ( 0 ; i_m ),     for  m = 0,1,2,3.
    e.g., i₃ = ( i₃ ; 0 ) ,  and i₆ = ( 0 ; i₂ ) .
 Then by Cayley-Dickson process the multiplication rule can be found,
for example,   i₃ i₆= ( 0 ; i₂ i₃) = ( 0 ; i₁ ) = i₅, and summarized in the following seven cycles:
(1,2,3) , (1,4,5) , (1,7,6) , (2,4,6) , (2,5,7) , (3,4,7) , (3,6,5) .            ..... details