On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_{\mathrm{\infty }}$-algebras

In this article, the Virasoro-type reduction and the corresponding inverse reductions are established for W-algebras associated with classical Lie type and nilpotent orbits of height two. Moreover, these results are lifted to the universal objects by analyzing the Virasoro-type reduction of the vertex algebra ${\mathcal{W}}_{\mathrm{\infty }}^{\mathfrak{s}\mathfrak{p}}$.