bi-gyrogroup: the group-like structure induced by bi-decomposition of groups

He decomposition γ=bh of a group γ into a subset b and a subgroup h of γ induces, under general conditions, a group-like structure for b, known as a gyrogroup. the famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by the einstein velocity addition law of special relativity theory. the latter leads to the lorentz transformation group so(1,n), n∈n, in pseudo-euclidean spaces of signature (1,n). the study in this article is motivated by generalized lorentz groups so(m,n), m,n∈n, in pseudo-euclidean spaces of signature (m,n). accordingly, this article explores the bi-decomposition γ=hlbhr of a group γ into a subset b and subgroups hl and hr of γ, along with the novel bi-gyrogroup structure of b induced by the bi-decomposition of γ. as an example, we show by methods of clifford algebras that the quotient group of the spin group spin(m,n) possesses the bi-decomposition structure.