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The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space ofthe vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is theMoonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and$V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodularlattice~$\textrm{II}_{1,1}$. We construct vertex algebra elements that projectto bases for subalgebras of~$\mathfrak m$ isomorphic to~$\mathfrak{gl}_{2}$,corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Ourmethod requires the existence of pairs of primary vectors in $V^{\natural}$satisfying some natural conditions, which we prove. We show that the action of$\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an$\M$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to afixed imaginary simple root. We use the generating function for dimensions ofsubspaces of primary vectors of $V^\natural$ to prove that this action isnon-trivial for small values of $j$.

Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra