The symmetric invariants of centralizers and Slodowy grading II

Let $mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $ell$ over an algebraically closed field $Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $mathfrak{sl}_2$-triple of g. Denote by $mathfrak{g}^{e}$ the centralizer of $e$ in $mathfrak{g}$ and by ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ the algebra of symmetric invariants of $mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $ell$ homogenous elements of ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ in $(mathfrak{g}^{e})^{*}$ has codimension $ell$. If $e$ is good then ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that $e$ is good if and only if for some homogenous generating sequence $q_1,ldots,q_ell$, the initial homogenous components of their restrictions to $e+mathfrak{g}^{f}$ are algebraically independent over $Bbbk$.