  Let $\mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}})$ be the universal
$\mathcal{W}$-algebra associated to $\mathfrak{sl}_4$ with its subregular
nilpotent 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 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.
