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 [Nijenhuis, 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?

[Nijenhuis, Richardson] Nijenhuis, Richardson - Deformations of Lie algebra structures

[Nijenhuis, Richardson 1] Nijenhuis, Richardson - Cohomology and deformations in Graded Lie algebras

[Nijenhuis, Richardson 2] Nijenhuis, Richardson - Cohomology and deformations of algebraic structure