TITLE: System of weights for nilpotent Lie algebras
QUESTION [5 upvotes]: I am studying nilpotent Lie algebra theory. The subject is really new to me and I am studying by myself. I'd love your help with this.
Let $\mathfrak{n}$ be a finite-dimensional nilpotent Lie algebra (over an algebraically closed field of characteristic zero) and let $\operatorname{Der}(\mathfrak{n})$ be the algebra of derivations of $\mathfrak{n}$. The system of weights of $\mathfrak{n}$ is defined as being that of the natural representation of a "maximal torus" $T$ in $\operatorname{Der}(\mathfrak{n})$ and the $\operatorname{rank}$ is the dimension of $T$. By remarkable result due to Gabriel Favre (see [F]), it is known that for a fixed integer $n$ there are finitely systems of weights. Let $T$ be a system of weights, we denote by $\mathrm{N}(T)$ the class of those Lie algebras having the system of weights $T$.
My questions are:

For a fixed integer $n$, are these system of weights classified?
For a fixed integer $n$, can rank-one system of weights explicitly written?
Are classified rank-one system of weights $T$ such that $\sharp\mathrm{N}(T)=1$
Is there a good book or resource for learning about this topic and in general, about nilpotent Lie algebras (over $\mathbb{C}$ or $\mathbb{R}$)?

Any help is much appreciated!
[F] Favre, G.: Système de poids sur une algèbre de Lie nilpotente. Manuscripta Math. 9 (1973), 53-90.

REPLY [4 votes]: 1.) No, the weight systems are not classified in general. However, the weight systems for complex nilpotent Lie algebras of dimension $n\le 7$ have been computed by Roger Carles, in  "Weight systems for complex nilpotent Lie algebras and application to the varieties of Lie algebras", in $1996$. Another reference, in addition to Pasha's references, is this paper of Magnin of $1998$.
2.), 3.) I think no.
4.) There are several books (e.g., by Goze, Onischik, Vinberg, etc.). Also, several articles discuss nilpotent Lie algebras in general, e.g. the article by E. M. Luks, What is a typical nilpotent Lie algebra ? For faithful representations of nilpotent Lie algebras, see, for example, here.