Cosets of the ${\mathcal{W}}^k({\mathfrak{s}\mathfrak{l}}_4,f_{\text{subreg}})$-algebra

Let ${\mathcal{W}}^k({\mathfrak{s}\mathfrak{l}}_4,f_{\text{subreg}})$ be the universal $\mathcal{W}$-algebra associated to ${\mathfrak{s}\mathfrak{l}}_4$ with its subregular nilpotent element, and let $\mathcal {W}_k(\mathfrak{sl}4, f{\text {subreg}})$beitssimplequotient\mathrm{.}ThereisaHeisenbergsubalgebra$$\mathcal{H}$$,andwedenoteby$$\mathcal{C}$^k$thecoset$$\text{Com}$($\mathcal{H}$, \mathcal {W}^k(\mathfrak{sl}4, f{\text {subreg}}))$, and by ${\mathcal{C}}_k$ its simple quotient. We show that for $k=-4+(m+4)\mathrm{/}3$ where $m$ is an integer greater than $2$ and $m+1$ is coprime to $3$, ${\mathcal{C}}_k$ is isomorphic to a rational, regular $\mathcal{W}$-algebra $\mathcal{W}({\mathfrak{s}\mathfrak{l}}_m,f_{\text{reg}})$. In particular, ${\mathcal{W}}_k({\mathfrak{s}\mathfrak{l}}_4,f_{\text{subreg}})$ is a simple current extension of the tensor product of $$\mathcal{W}$(\mathfrak{sl}m, f{$\text{reg}$})$ with a rank one lattice vertex operator algebra, and hence is rational.