S := ( A ; B ) .

By Cayley-Dickson process the multiplication of two sedenions, S and T, in terms of four octonions, A, B, C, D, may be defined as
        S T := ( A ; B ) ( C ; D ) := ( AC + gD*B ; BC* + DA ) ,

where B* is a conjugate of B, and g is a field parameter.
 A sedenion, S, may also be written componentwise as
        S := _m=0S¹⁵  s_m e_m    =  s₀ e₀  +  _j=1S¹⁵  s_j e_j ,

where e_m are the basal elements of S.  The basal elements, e_m ,  may be constructed from octonion basal elements, i_m , analogous to the method used for octonion basal elements.  If we choose the multiplication rule for octonions as described for our octonion basis   ( i.e. the field parameter, g, is chosen as -1 at each iteration of the Cayley-Dickson process ) we obtain the multiplication rule summarized in the following table.
 

Just as with octonions one can write the multiplication rule in somewhat more compact form by means of the following 35 sedenion cycles:

        ( 1,2,3 ), ( 1,4,5 ), ( 1,7,6 ), ( 1,8,9 ), ( 1,11,10 ), ( 1,13,12 ), ( 1,14,15 )
        ( 2,4,6 ), ( 2,5,7 ), ( 2,8,10 ), ( 2,9,11 ), ( 2,14,12 ), ( 2,15,13 ),
        ( 3,4,7 ), ( 3,6,5 ), ( 3,8,11 ), ( 3,10,9 ), ( 3,13,14 ), ( 3,15,12 ),
        ( 4,8,12 ), ( 4,9,13 ), ( 4,10,14 ), ( 4,11,15 ),
        ( 5,8,13 ), ( 5,10,15 ), ( 5,12,9 ), ( 5,14,11 ),
        ( 6,8,14 ), ( 6,11,13 ), ( 6,12,10 ), ( 6,15,9 ),
        ( 7,8,15 ), ( 7,9,14 ), ( 7,12,11 ), ( 7,13,10 ).

The multiplication of two sedenion basal elements, e_j   and e_k  , is given by

      e_j   e_k    =  -  d_{j k}  + _(jkm)S e _jkm e _m  ,    ( j, k, m = 1,2,...,15 )   ,

where the summation is over all possible permutations of the sedenion cycle, (j,k,m), and the structure constant,  e_jkm  , is totally anti-symmetric in its indices and is given by
 

e jkm   = {

1   if  (j,k,m) is an even permutation, - 1   if  (j,k,m) is an odd permutation,    0   otherwise,

}  of a sedenion cycle,

and as usual,
 

dj k   =  {

1   if j = k 0   otherwise.

As an example, the sedenion cycle, ( 5,12,9 ) implies that
    e₉   e₁₂  =   - e₅ ,    e₉   e₅  =    e₁₂  ,  etc..