Journal of Lie Theory

We prove that every $2$-solvable Frobenius Lie algebra splits as a semidirect sum of an $n$-dimensional vector space $V$ and an $n$-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of $V$. We supply a complete classification of $2$-solvable Frobenius Lie algebras corresponding to nonderogatory endomorphisms, as well as those given by maximal Abelian nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature $(n\!-\!1,0,1)$. In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general up to dimension $8$. We correct and complete the classification list of MASAs of $\mathfrak{sl}(4,\mathbb{R})$ by Winternitz and Zassenhaus. As a biproduct, we give a simple proof that every nonderogatory endormorphism of a real vector space admits a Jordan form and also provide a new characterization of Cartan subalgebras of $\mathfrak{sl}(n,\mathbb{R})$.