| Summary: | The 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.
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