Graph criterion for semisimplicity of Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional graph-admissible Lie algebra associated with the labeled directed graph $G(V,E)$. Graph semisimplicity conjecture. The Lie algebra $\mathfrak{g}$ is semisimple if and only if every vertex is part of a closed directed walk that induces a self-contained subgraph of $G(V,E)$. This proposes a graph-theoretic characterization of semisimplicity; the source presents it as a strong condition motivated by preceding graph criteria.