Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras, and almost complex product structures

We study Lagrangian and orthogonal splittings\textbf{\ }of quadratic vector spaces establishing an equivalence with complex product structures. Then we show that a Manin triple equipped generalized metric $\mathcal{G}+% \mathcal{B}$ such $\mathcal{B}$ is $\mathcal{O}$-operator extension $\mathcal{G}$ of mass -1 can be turned in another admits also splitting in\textbf{\ }Lie ideals. Conversely, Lie algebra direct sum pair anti-isomorphic algebras, after similar steps as the previous case, admitting into