Free nilpotent and nilpotent quadratic Lie algebras

In this paper we introduce an equivalence between the category of the t-nilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms over the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphisms, and over any field of characteristic zero), in the following cases: d=2 and t≤5, d=3 and t≤3.