  Octonions & Sedenions
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Basis for Octonions:
To construct basal elements of octonion from that of quaternions, let
ir  ( r = 0,1,2,3 ) denote quaternion basis, and
em  ( m = 0,1,...,7 ) denote octonion basis.
Let    em  :=  ( im ; 0 ) ,     for m  =  0,1,...,3,      and
em  :=  ( 0 ; im-4 )  ,  for m  =  4,...,7.
e.g.    e0  =  ( i0 ; 0 ) = ( 1 ; 0 ) = 1,
e2 = ( i2  ; 0 ) ,            e7 = ( 0 ; i3 ) .
Then the multiplication rule for the octonion basal elements can be found using Cayley-Dickson process (when  m is chosen as -1 at each iteration).   For example,
e4 e5 = ( 0 ; 1 ) ( 0 ; i1 ) = ( - i1* ; 0 ) = ( i1 ; 0 ) = e1  .

A multiplication table can then be constructed:

 e0 = 1 e1 e2 e3 e4 e5 e6 e7 e0 = 1 1 e1 e2 e3 e4 e5 e6 e7 e1 e1 -1 e3 -e2 e5 -e4 -e7 e6 e2 e2 -e3 -1 e1 e6 e7 -e4 -e5 e3 e3 e2 -e1 -1 e7 -e6 e5 -e4 e4 e4 -e5 -e6 -e7 -1 e1 e2 e3 e5 e5 e4 -e7 e6 -e1 -1 -e3 e2 e6 e6 e7 e4 -e5 -e2 e3 -1 -e1 e7 e7 -e6 e5 e4 -e3 -e2 e1 -1

The multiplication rule for the pure octonion basis ( ej , j = 1,2,...,7 ) can be represented diagrammatically by

where the arrows indicate the signs of the resulting basal elements.
e.g.,    e2 i7  =  - e5 ,    e3 e4  =  e7  .

Another more text friendly way to express the multiplication rule is by means of octonion cycles.   The multiplication of 2 octonion basal elements, ej   and ek , is given by
ej   ek    =  -  dj k  + (jkm)S e jkm e m  ,    ( j, k, m = 1,2,...,7 )   ,
where the summation is over all possible permutation of (j,k,m), and the structure constant  ejkm is totally anti-symmetric in its indices whose value is determined from the following octonion cycles:
( 1,2,3 ),  ( 1,4,5 ),  ( 1,7,6 ),  ( 2,4,6 ),  ( 2,5,7 ),  ( 3,4,7 ),  ( 3,6,5 ) ,

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 an octonion cycle,

and as usual

 dj k   =  { 1   if j = k 0   otherwise.

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