The Monster Lie algebra $\mathfrak{m}$ is a quotient of the physical space of the vertex algebra $V^\natural\otimes V_{1,1}$ where $V^\natural$ is the Moonshine module of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is a vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. It is known that $\mathfrak{m}$ has $\mathfrak{gl}_2$-subalgebras generated by both real and imaginary root vectors and that the Monster finite simple group $\mathbb{M}$ acts trivially on the $\mathfrak{gl}_2$-subalgebra corresponding to the unique real simple root. In this talk, I will discuss the construction of families of elements in $V^\natural\otimes V_{1,1}$ that can be used to construct $\mathfrak{gl}_2$-subalgebras corresponding to imaginary simple roots $(1,n)$ of $\mathfrak{m}$. The action of $\mathbb{M}$ on $V^\natural$ induces an $\mathbb{M}$-action on the Chevalley generators of these $\mathfrak{gl}_2$-subalgebras in $\mathfrak{m}$. It remains to determine if this $\mathbb{M}$-action is nontrivial. This talk is based on joint work with Lisa Carbone, Elizabeth Jurisich, Maryam Khaqan, and Scott H. Murray.