Is it true that $\mathfrak{g}=\mathfrak{g}_e\oplus[x,\mathfrak{g}]$? Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $(e,f,h)$ a principal $\mathfrak{sl}_2$-triple (see below). Let $\mathfrak{g}_e$ be the centralizer of $e$ and let $x\in f+\mathfrak{g}_e$. Is it true that
$$\mathfrak{g}=\mathfrak{g}_e\oplus[x,\mathfrak{g}]?$$
Remarks. 
(i) The affine space $f+\mathfrak{g}_e$ is sometimes called the Kostant section, Kostant slice, or principal Slodowy slice. We have $f+\mathfrak{g}_e\subseteq\mathfrak{g}^{reg}$.
(ii) Since $x,e\in\mathfrak{g}^{reg}$ we have $\dim\mathfrak{g}_e+\dim[x,\mathfrak{g}]=\dim\mathfrak{g}$ so it suffices to show that $\mathfrak{g}=\mathfrak{g}_e+[x,\mathfrak{g}]$.
(iii) When $\mathfrak{g}$ is of type $A_1$ or $A_2$, I verified by brute force that indeed $\mathfrak{g}=\mathfrak{g}_e\oplus[x,\mathfrak{g}]$.

Definition. A principal $\mathfrak{sl}_2$-triple in $\mathfrak{g}$ is a triple $(e,f,h)$ of elements of $\mathfrak{g}$ such that $[h,e]=2e,[h,f]=-2f,[e,f]=h$ and $e$ is regular, i.e. its centralizer has dimension equal to the rank of $\mathfrak{g}$.
 A: This is indeed the case. To see this, filter $\mathfrak{g}$ by $Ker(ad_e^i)$. 
Let $Gr(\mathfrak{g})$ be the associated graded. We have a symbol map 
$sym : \mathfrak{g} \to Gr(\mathfrak{g})$ given by sending an element to the "leading term" w.r.t. this filtration. 
It would suffice to show that it is generated by the symbols of elements of $\mathfrak{g}_e + [x,\mathfrak{g}]$, by a standard descent argument (look at the least element not in your subspace in terms of this filtration) 
But, after passage to the associated graded, $ad_x$ acts on symbols like $ad_f$ modulo previous graded pieces. 
More precisely, we have $sym(ad_x(u)) = sym(ad_f(u))$ for every $u$. 
This shows that the image of the symbol don't change if you replace $x$ by $f$, so we can reduce to this case. But for $x = f$ the statement is a standard result on representations of $\mathfrak{sl}_2$ with no trivial components (which follows from the triple being principal in this case).   
Edit: As mensioned in the comment below, the condition on non-triviality of the representation is irrelevant here. It is just true for every $\mathfrak{sl}_2$ representation. 
