  The equivariant $\mathcal{W}$-algebra of a simple Lie algebra $\mathfrak{g}$
is a BRST reduction of the algebra of chiral differential operators on the Lie
group of $\mathfrak{g}$. We construct a family of vertex algebras
$A[\mathfrak{g}, \kappa, n]$ as subalgebras of the equivariant
$\mathcal{W}$-algebra of $\mathfrak{g}$ tensored with the integrable affine
vertex algebra $L_n(\check{\mathfrak{g}})$ of the Langlands dual Lie algebra
$\check{\mathfrak{g}}$ at level $n\in \mathbb{Z}_{>0}$. They are conformal
extensions of the tensor product of an affine vertex algebra and the principal
$\mathcal{W}$-algebra whose levels satisfy a specific relation.
