Historically, the question of how far the properties of the subgroup lattice of a group G determine those of G has been extensively studied. Similar questions have been formulated for Lie algebras and restricted Lie algebras.
In this talk, we will give a general characterization for restricted Lie algebras having a distributive lattice of restricted subalgebras, and also show that this result takes particularly satisfying forms if we impose additional conditions to the base field or to the algebra.
Also, we will give an overview of restricted Lie algebras whose lattices of restricted subalgebras satisfy other properties as being Boolean or modular.

This talk is based on joint works with Nicola Maletesta, Salvatore Siciliano and David Towers.