  The algebra of octonions is non-associative (as well as non-commutative).
This makes it very difficult to derive algebraic results, and to perform
computation with octonions. Given a product of more than two octonions, in
general, the order of evaluation of the product (placement of parentheses)
affects the result. Inspired by the concept of the commutator $[x,y]=
x^{-1}y^{-1}xy$ we show that an associator can be defined that multiplies the
result from one evaluation order to give the result from a different evaluation
order. For example, for the case of three arbitrary octonions $x$, $y$ and $z$
we have $((xy)z)a = x(yz)$, where $a$ is the associator in this case. For
completeness, we include other definitions of the commutator, $[x,y]=xy-yx$ and
associator $[x,y,z]=(xy)z-x(yz)$, which are well known, although not
particularly useful as algebraic tools. We conclude the paper by showing how to
extend the concept of the multiplicative associator to products of four or more
octonions, where the number of evaluation orders is greater than two.
