Infinite dimensional analogues of nilpotent and solvable Lie algebras

We study infinite-dimensional analogues of nilpotent and solvable Lie algebras, focusing on the classes of pro-nilpotent, residually nilpotent, pro-solvable and residually solvable Lie algebras. We extend classical triangularization results (Engel's and Lie's theorems) to the pro-setting and establish existence results for the pro-nilpotent radical in pro-solvable algebras and in certain residually solvable algebras. We adapt finite-dimensional construction methods to produce residually solvable extensions with a given pro-nilpotent radical under natural finiteness conditions. By analyzing derivations and maximal tori of pro-nilpotent algebras, we extend the notion of rank and show that, for pro-nilpotent algebras of maximal rank, every derivation of a maximal residually solvable extension is inner. Finally, we describe standard constructions (tensor and direct sum products, central extensions) that preserve pro-nilpotency.