Nonempty alternative quasigroup equals alternative loop

Statement
The following are equivalent for a quasigroup:


 * 1) It is nonempty and is an alternative magma under its multiplication.
 * 2) It is an alternative loop, i.e., it is an algebra loop and is an alternative magma under its multiplication.

Note that an algebra loop is a quasigroup with a two-sided neutral element, so the above simply says that for a nonempty quasigroup, alternativity guarantees the existence of a neutral element.

Related facts

 * Nonempty associative quasigroup equals group