Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is a well-known theorem (of Cartan for $R = \mathbb{C}$, and Chevalley in characteristic zero, and I think Humphreys in general) that all Cartan $R$-subalgebras of $\mathfrak{g}_R$ are $G(R)$-conjugate, but this fails for arbitrary $R$. For example, when $R$ is the field of real numbers and $G=\operatorname{SL}_2$, the Lie algebra of a non-split maximal torus is not $G(R)$-conjugate to $\mathfrak{t}_R$). Instead of this general conjugacy, I'd like to know how well the action of $G(R)$ approximates the action of the group of $R$-Lie algebra automorphisms of $\mathfrak{g}_R$.

**Question 1:** Does $G(R)$ act transitively on the $\operatorname{Aut}_{R\text{-Lie}} \mathfrak{g}_R$-orbit of $\mathfrak{t}_R$?

There is a group scheme analogue of this:

**Question 2:** Does $G(R)$ act transitively on the $\operatorname{Aut}_{R\text{-gp}.} G_R$-orbit of $T_R$?

It seems pretty clear that a positive answer to the first question implies a positive answer to the second, but I don't know if the reverse holds. As far as I can tell, the answer to both questions is positive when the canonical isogeny $Z(G) \times D(G) \to G$ is an isomorphism (that is, when $G$ is a direct product of a torus and an adjoint type group). The problem is that, while the adjoint actions of $G$ on $G_R$ and $\mathfrak{g}_R$ factor through the quotient $G/Z(G)$ by the center, the scheme-theoretic surjection $G \to G/Z(G)$ may not yield a surjection on rational points! There is a cokernel that lies in a flat $H^1$ group. Even so, I have not been able to construct an example giving a negative answer to either question.

Naturally, I would appreciate any insights even in the case of fields, or with restricted characteristic.

**Edit 9 hours later:** L. Spice has helpfully pointed out in a comment that when $R$ is a field, then the answer to question 2 is "yes". I now see that this is theorem 20.9(ii) in Borel's *Linear algebraic groups*. I haven't gone through the details yet, but it looks like the proof in the text can be extended to conjugacy that is Zariski local in $R$.

Question 2in that case is 'yes'. $\endgroup$7more comments