  Octonions & Sedenions

..

SIMPLE TOOLBOX

Here are simple tools for expressing some of the basic algebraic properties of octonions, namely the commutator, the associator, and the two structure constants.

Let A,B,C denote three octonions.

(1) Commutator of A and B is defined as
[ A, B ] :=  AB - BA  {
 = 0 if A and B commute not 0 if A and B do not commute.

(2) Associator of A, B, C is defined as
( A, B, C ) :=  (AB) C - A (BC)
 = 0 if associative not 0 if not associative.

Using the associator the alternative property of octonions is expressed as

( A, A, B ) = (AA) B - A (AB) = 0   .

We often work with the basis of octonions. i.e an octonion, A, may be expressed in terms of the basis, i , as
A := m=0S7  am im  =  a0 i0  +  j=1S7  aj ij     ,

where Greek indices represent 0,1,...,7,  and Latin indices represent 1,...,7.
Here, the non-commutative and non-associative properties of basal elements are expressed, more explicitly, using structure constants, emnr , and  emnrs ,  respectively.    Let im , and in  be two basal elements of octonions.

(3) Structure constant ( 1st kind, relating to commutators ), emnr , is defined via a product of two basal elements,
[ im , in ] :=  im in   -  in im  =  2 e mnr  i  .
e mnr  are real numbers which equals 0 only when im  commutes with in  .

(4) Structure constant ( 2nd kind, relating to associators ), emnrs , is defined via a product of three basal elements,
( im  , in ,  ir  )  :=  ( im  in  )  ir    -   im  ( in    ir  )  =  2  emnrs i  .
emnrs  are real numbers.

...