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.



Chevalley restriction theorem for non-split Cartan
Taking invariants commutes with flat base change, so, in particular, with extensions of the base field. This implies that Chevalley's restriction theorem works over any field of characteristic zero. Also Shephard-Todd works over any field since a polynomial ring with a positive grading has only trivial forms. To see this take any $k$-basis of $\mathfrak m/\mathfrak m^2$ (with $\mathfrak m$ being the maximal homogeneous ideal) and lift it to homogeneous ring elements which are defined over $k$. These will freely generate the ring.


