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_m  , as 
     A := _m=0S⁷  a_m i_m  =  a₀ i₀  +  _j=1S⁷  a_j i_j     , 

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, e_mnr , and  e_mnrs ,  respectively.    Let i_m , and i_n  be two basal elements of octonions.

(3) Structure constant ( 1st kind, relating to commutators ), e_mnr , is defined via a product of two basal elements,
     [ i_m , i_n ] :=  i_m i_n   -  i_n i_m  =  2 e _mnr  i_r   .
e _mnr  are real numbers which equals 0 only when i_m  commutes with i_n  .

(4) Structure constant ( 2nd kind, relating to associators ), e_mnrs , is defined via a product of three basal elements,
     ( i_m  , i_n ,  i_r  )  :=  ( i_m  i_n  )  i_r    -   i_m  ( i_n    i_r  )  =  2  e_mnrs i_s   .
e_mnrs  are real numbers.