On maximal solvable extensions of nilpotent Lie algebras

In this paper, we provide a complete description of complex maximal solvable extensions for a certain class of nilpotent Lie algebras. In particular, we show that, up to isomorphism, a solvable extension of a $d$-locally diagonalizable nilpotent Lie algebra is unique and is realized as the semidirect product of its nilradical with a maximal torus. This result resolves a conjecture of \v{S}nobl concerning the uniqueness of maximal solvable extensions under the condition $d$-locally diagonalizability on the nilradical. Moreover, we extend this description to the setting of Lie superalgebras and present an alternative method for constructing such maximal solvable extensions. Finally, we discuss further aspects and open questions related to maximal solvable extensions of nilpotent Lie algebras