  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.
