Left distributive quasigroups and gyrogroups (Q2743895)

The following propositions are proved: every quasigroup of reflection is isotopic to some gyrocommutative gyrogroup (i.e. a left Bol loop with automorphic inverse property); every left distributive quasigroup satisfying some specific condition is isotopic to some nongyrocommutative gyrogroup (i.e. a homogeneous loop with the left loop property). Some properties of local gyrocommutative gyrogroups are described in terms of reductive homogeneous spaces and semidirect products.