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Let $\mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}})$ be the universal$\mathcal{W}$-algebra associated to $\mathfrak{sl}_4$ with its subregularnilpotent element, and let $\mathcal {W}_k(\mathfrak{sl}_4, f_{\text{subreg}})$ be its simple quotient. There is a Heisenberg subalgebra$\mathcal{H}$, and we denote by $\mathcal{C}^k$ the coset$\text{Com}(\mathcal{H}, \mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}}))$,and by $\mathcal{C}_k$ its simple quotient. We show that for $k=-4+(m+4)/3$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$\mathcal{W}(\mathfrak{sl}_m, f_{\text{reg}})$. In particular,$\mathcal{W}_k(\mathfrak{sl}_4, f_{\text {subreg}})$ is a simple currentextension of the tensor product of $\mathcal{W}(\mathfrak{sl}_m,f_{\text{reg}})$ with a rank one lattice vertex operator algebra, and hence isrational.

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