An arbitrary section of the canonical projection of a group onto the cosets modulo a subgroup is associated with a binary operation on the cosets. We provide sufficient conditions for obtaining a left loop, a left gyrogroup or a gyrocommutative gyrogroup in such a way. The non-positively curved sections in Lie groups allow a scalar multiplication, which turns them into quasi left Lie gyrovector spaces. The left invariant metrics on homogeneous spaces turn out to be compatible with the gyro-structure. For instance, their geodesics are gyro-lines; the associated distance to the origin is a gyro-homogeneous norm, satisfying gyro-triangle inequality; etc. The work establishes infinitesimal criteria for a homogeneous space to bear a left Lie gyrovector space or a Lie gyrovector space structure. It characterizes the Cartan gyrovector spaces and works out explicitly the example of the upper half-plane.
"Lie Gyrovector Spaces." J. Geom. Symmetry Phys. 1 3 - 53, 2004. https://doi.org/10.7546/jgsp-1-2004-3-53