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$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups, ...

First please see this very simple fact: Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace. By an invariant subspace we mean a space $M$ ...