TITLE: Lie algebras with trivial second cohomology group
QUESTION [8 upvotes]: Let $\mathfrak{L}_n(\mathbf{C})$ be the set of all $n$-dimensional complex Lie algebras. One know that in $\mathfrak{L}_n(\mathbf{C})$ there exist only a finite number of Lie algebras with trivial second cohomology group with coefficients in the adjoint representation (by using a rigidity argument of Nijenhuis and Richardson). 
Question: Fixing $n$, are Lie algebras $\mathfrak{g} \in \mathfrak{L}_n(\mathbf{C})$ such that $H^2(\mathfrak{g},\mathfrak{g})=0$ classified?
References: 

Nijenhuis, Richardson - Deformations of Lie algebra structures
Nijenhuis, Richardson - Cohomology and deformations in Graded Lie algebras
Nijenhuis, Richardson - Cohomology and deformations of algebraic structure

REPLY [4 votes]: Even if we consider more generally cohomologically rigid Lie algebras $L$, which satisfy $H^i(L,L)$ for all $i\ge 0$, a classification is not possible.
Roger Carles proved in 1985 in his paper "Sur certain classes d'algebres de Lie rigides", that every complete Lie algebra $L$ with abelian nilradical ${\rm nil}(L)$ is cohomologically rigid. He proves a lower bound of the number $s_n$ of non-isomorphic solvable cohomologically rigid Lie algebras of dimension $n$, which is indeed
$$
\Gamma(\sqrt{n})\le s_n,
$$
for all $n\ge 81$, with the Gamma function. This shows that a classification is more or less hopeless.