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The coset (commutant) construction is a fundamental tool to construct vertexoperator algebras from known vertex operator algebras. The aim of this paper isto provide a fundamental example of the commutants of vertex algebras ousidevertex operator algebras. Namely, the commutant $C$ of the principal subalgebra$W$ of the $A_1$ lattice vertex operator algebra $V_{A_1}$ is investigated. Anexplicit minimal set of generators of $C$, which consists of infinitely manyelements and strongly generates $C$, is introduced. It implies that the algebra$C$ is not finitely generated. Furthermore, Zhu's Poisson algebra of $C$ isshown to be isomorphic to an infinite-dimensional algebra$\mathbb{C}[x_1,x_2,\ldots]/(x_ix_j\,|\,i,j=1,2,\ldots)$. In particular, theassociated variety of $C$ consists of a point. Moreover, $W$ and $C$ areverified to form a dual pair in $V_{A_1}$.

On the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra