Double extension of flat pseudo-Riemannian F-Lie algebras

We define the concept of a flat pseudo-Riemannian F-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent F-Lie algebras possessing one-dimensional light-cone subspaces. A similar result can be established for nilpotent Lie algebras equipped with flat scalar products of signature $(2,n-2)$ where $n\ge 4$. Furthermore, we use this technique to construct Poisson algebras exhibiting compatibility with flat scalar products.