O(4,4) dualities and Manin triples

We provide a coarse classification of all 8-dimensional Manin triples, that describe Poisson–Lie T-dualities between 4-dimensional group manifold solutions to supergravity equations. We find several such dualities and one Poisson–Lie triality. For each class we refine the classification by marking whether the corresponding Drinfeld double can be obtained by a Yang–Baxter deformation of the initial algebra paired by the four-dimensional Abelian algebra.