Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends by derivations to a representation of $\mathfrak{g}$ in $S(\mathfrak{g})$. We identify $S(\mathfrak{g})$ and $\mathbb{C}[\mathfrak{g}]$ as $\mathfrak{g}$-modules via the Killing form. The most famous description of $S(\mathfrak{g})^{\mathfrak{g}}$ goes as follows: the Chevalley restriction theorem states that the restriction map $\mathbb{C}[\mathfrak{g}]^\mathfrak{g} \rightarrow \mathbb{C}[\mathfrak{h}]^W$ is an isomorphism and the Chevalley-Sheppard-Todd theorem implies that the Weyl group invariants are a polynomial ring on $\operatorname{rank}(\mathfrak{g})$ generators.

I am interested in another approach to constructing the invariant rings via restriction. Take an $n$-dimensional vector space $V$ and set $\mathfrak{g} = \mathfrak{gl}(V)$. The characteristic polynomial $p(t, x)$ of an element $x \in \mathfrak{g}$ can be written as $t^n + \sum_{i=0}^{n-1} p_i(x) t^i$, and the coefficients $p_i$ form a complete set of algebraically independent generators for $S(\mathfrak{g})^{\mathfrak{g}}$. Now suppose that $\mathfrak{k} \subseteq \mathfrak{g}$ is a classical Lie algebra of type $\mathfrak{so}(V)$ or $\mathfrak{sp}(V)$. We can consider the restrictions $p_{i}\vert_{\mathfrak{k}}$ and in this case we have $p_{2i}\vert_{\mathfrak{k}} = 0$ for all $i$ and the remaining invariants form a complete set of algebraically independent generators for $S(\mathfrak{k})^\mathfrak{k}$, unless of course $\mathfrak{k}$ has type ${\sf D}$, in which case $p_{n-1} = P^2$ for some $P \in S(\mathfrak{g})$ known as the Pfaffian. In this case $p_{1}\vert_{\mathfrak{k}}, p_{3}\vert_{\mathfrak{k}},...,p_{n-3}\vert_{\mathfrak{k}}, P$ form a basic set of generators for the symmetric invariants. This is all very classical and probably goes back to Weyl.

What I would like to know is whether we can play the same game with exceptional Lie algebras. This is my question: for $\mathfrak{g}$ simple and exceptional, does there always exist a simple finite dimensional representation $\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ such that some of the restrictions $p_0\vert_{\mathfrak{g}},...,p_{n-1}\vert_{\mathfrak{g}}$ are zero, whilst the others form a complete set of algebraically independent generators for $S(\mathfrak{g})^{\mathfrak{g}}$, perhaps after Pfaffing around with them a bit?