The Paige loop of a field is defined as follows:
- Let be the set of invertible elements in the split octonion algebra over , made into a Moufang loop by the multiplication in the algebra.
- Let be the center of . can also be defined as the subgroup of comprising . Note that has size one if has characteristic two and size two otherwise.
- The Paige loop of is defined as the quotient loop.
The Paige loop of any field is a simple Moufang loop, and it is always non-commutative and non-associative, so Paige loops are examples of simple Moufang loops that are not simple groups. For finite orders, these are the only examples of simple Moufang loops that are not simple groups.
The order (i.e., size) of the Paige loop over a finite field of size is given by: