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

The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset $B$ and a subgroup $H$ of $\Gamma$ 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 $\mathrm{SO}(1,n)$, $n\in\mathbb{N}$, in pseudo-Euclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $\mathrm{SO}(m, n)$, $m, n\in\mathbb{N}$, in pseudo-Euclidean spaces of signature $(m, n)$. Accordingly, this article explores the bi-decomposition $\Gamma = H_LBH_R$ of a group $\Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of $\Gamma$, along with the novel bi-gyrogroup structure of $B$ induced by the bi-decomposition of $\Gamma$. As an example, we show by methods of Clifford algebras that the quotient group of the spin group $\mathrm{spin}(m, n)$ possesses the bi-decomposition structure.