  Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ and $G$ be the
corresponding simply connected algebraic group. Consider a nilpotent element
$e\in \mathfrak{g}$, the corresponding element $\chi=(e, \bullet)$ in
$\mathfrak{g}^*$, and the coadjoint orbit $\mathbb{O}=G\chi$. We are interested
in the set $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$ of codimension $1$ ideals
$J\subset \mathcal{W}$ in a finite $W$-algebra $\mathcal{W}=U(\mathfrak{g},
e)$. We have a natural action of the component group
$\Gamma=Z_G(\chi)/Z_G^\circ(\chi)$ on
$\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$. Denote the set of $\Gamma$-stable
points of $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$ by
$\mathfrak{J}\mathfrak{d}^{1}(\mathcal{W})^{\Gamma}$. For a classical
$\mathfrak{g}$ Premet and Topley proved that
$\mathfrak{J}\mathfrak{d}^{1}(\mathcal{W})^{\Gamma}$ is isomorphic to an affine
space. In this paper we will give an easier and shorter proof of this fact.
