Totally geodesic Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{S}\mathrm{L}(2,\mathbb{R})\times \mathrm{S}\mathrm{L}(2,\mathbb{R})$

In this paper, we study Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{S}\mathrm{L}(2,\mathbb{R})\times \mathrm{S}\mathrm{L}(2,\mathbb{R})$. First, we show that they split into four different classes depending on their behaviour with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.