Odd nilpotent element and $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$-subalgebra in $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$

In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$ lies inside an $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$-subalgebra. In the case of the classical Lie algebra ${\mathfrak{g}\mathfrak{l}}_m$, every nilpotent element can be embedded into an ${\mathfrak{s}\mathfrak{l}}_2$-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$, we define super Jordan matrices and prove that an odd nilpotent element $e$ is contained in an $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$-subalgebra if and only if $e$ lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size.