Classification of Lie algebras constructed from $\mathfrak{gl}_{m|n}$ via Derived Bracket

Derived brackets provide a mechanism for generating algebraic structures from graded Lie superalgebras, with applications in Poisson geometry, mathematical physics, and the theory of algebroids. In this paper, we present a complete structural and isomorphism classification of a family of Lie algebras constructed from the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ over a field $\mathbb{K}$ of characteristic zero via the derived bracket generated by an odd element $B$ satisfying $B^2 = 0$, which endows $\mathfrak{g}_{-1}$ with a Lie algebra structure denoted $\mathfrak{g}_{-1}^{B}$. We prove that for fixed dimensions $m$ and $n$, the isomorphism type of $\mathfrak{g}_{-1}^{B}$ is entirely determined by $r=\operatorname{rank}(B)$. In arbitrary dimensions, two such algebras are isomorphic if and only if they share the same rank $r$ and satisfy $\{m,n\}=\{p,q\}$. We explicitly compute the Levi-Malcev decomposition, proving the semisimple Levi factor is isomorphic to $\mathfrak{sl}(r)$, and provide exact formulas for the solvable radical and center.