Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of Chevalley which says that $k[\mathfrak{g}]^G\simeq k[\mathfrak{h}]^W$. A theorem of Chevalley–Shephard–Todd then states that this is a polynomial algebra.

I'm wondering to what extent these theorems remain true if $k$ is not algebraically closed. The case I'm actually interested in is when $k=\mathbb{C}((t))$ in which case, the conjugacy class of Cartan subalgebras are in bijection with conjugacy classes in the absolute Weyl group (i.e., the Weyl group over the algebraic closure).

Most references I know assume $k$ is algebraically closed or that $\mathfrak{h}$ is split. References dealing with non-split Cartan would be appreciated.

absolutelysimple and simply connected. One can thereby reduce to the case of absolutely simple $G$. Since $k$ is perfect with cohomological dimension 1, by a theorem of Steinberg $G$ is quasi-split (so only types A, D, and E$_6$ can be non-split). A preferred $G(k)$-conjugacy class of maximal $k$-tori is those in Borel $k$-subgroups, which are "induced tori"; maybe using these simplifies things? $\endgroup$ – nfdc23 Jul 23 '16 at 3:28