Minimal W-algebras with non-admissible levels and intermediate Lie algebras

In \cite{Kawasetsu:2018irs}, Kawasetsu proved that the simple W-algebra associated with a minimal nilpotent element $W_k(\mathfrak{g},f_{\theta })$ is rational and $C_2$-cofinite for $\mathfrak{g}=D_4,E_6,E_7,E_8$ with non-admissible level $k=-h^{\vee }\mathrm{/}6$. In this paper, we study $W_k(\mathfrak{g},f_{\theta })$ algebra for $\mathfrak{g}=E_6,E_7,E_8$ with non-admissible level $k=-h^{\vee }\mathrm{/}6+1$. We determine all irreducible (Ramond twisted) modules, compute their characters and find coset constructions and Hecke operator interpretations. These W-algebras are closely related to intermediate Lie algebras and intermediate vertex subalgebras.