The equivariant
W
\mathcal W
-algebra of a simple Lie algebra
g
\mathfrak {g}
is a BRST reduction of the algebra of chiral differential operators on the Lie group of
g
\mathfrak {g}
. We construct a family of vertex algebras
A
[
g
,
κ
,
n
]
A[\mathfrak {g}, \kappa , n]
as subalgebras of the equivariant
W
\mathcal W
-algebra of
g
\mathfrak {g}
tensored with the integrable affine vertex algebra
L
n
(
g
ˇ
)
L_n(\check {\mathfrak {g}})
of the Langlands dual Lie algebra
g
ˇ
\check {\mathfrak {g}}
at level
n
∈
Z
>
0
n\in \mathbb {Z}_{>0}
. They are conformal extensions of the tensor product of an affine vertex algebra and the principal
W
\mathcal {W}
-algebra whose levels satisfy a specific relation.
When
g
\mathfrak {g}
is of type
A
D
E
ADE
, we identify
A
[
g
,
κ
,
1
]
A[\mathfrak {g}, \kappa , 1]
with the affine vertex algebra
V
κ
−
1
(
g
)
⊗
L
1
(
g
)
V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g})
, giving a new and efficient proof of the coset realization of the principal
W
\mathcal W
-algebras of type
A
D
E
ADE
.