The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of
the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the
Moonshine module of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the 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 simple group $\mathbb{M}$ acts trivially on the
$\mathfrak{gl}_2$-subalgebra corresponding to the unique real simple root. We
construct elements of $V$ that project under the quotient map to the
Serre-Chevalley generators of families of $\mathfrak{gl}_2$-subalgebras
corresponding to the imaginary simple roots $(1,n)$ of $\mathfrak m$ for
$0<n<100$. We prove the existence of primary vectors in $V^\natural$ of each
homogeneous weight $n$ and, for $0<n<100$, we show that there exist primary
vectors that can be used to construct the elements in $V$ corresponding to the
generators of our $\mathfrak{gl}_2$-subalgebras. We show that the action of
$\mathbb{M}$ on $V^\natural$ induces an action on the Serre-Chevalley
generators of the aforementioned subalgebras. We conjecture that this
$\mathbb{M}$-action is non-trivial.
