Octonion Basis

Octonions & Sedenions

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Basis for Octonions:
    To construct basal elements of octonion from that of quaternions, let
    i_r ( r = 0,1,2,3 ) denote quaternion basis, and
    em ( m = 0,1,...,7 ) denote octonion basis.
Let    e_m := ( i_m ; 0 ) ,     for m = 0,1,...,3,      and
         e_m := ( 0 ; i_m-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 ( e_j , 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, 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,...,7 )   ,
where the summation is over all possible permutation of (j,k,m), and the structure constant e_jkm 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

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