relative rank two group: structure of parabolic subgroup— high-level Jacobson--Morozov sl_2 triple

Given a parabolic subgroup $$P=MN$$ of a connected reductive group $$G$$ defined over a local field $$F$$, let $$W_M$$ be the relative Weyl group of $$M$$ in $$G$$, assume that the reduced roots relative to $$M$$ in $$G$$ form a root system of rank two, i.e. $$W_M=\left< S_\alpha, S_\beta \right>,$$ where $$\alpha$$ and $$\beta$$ are the simple roots of M in G, and $$S_\alpha$$ and $$S_\beta$$ are the corresponding simple reflections which satisfy $$S_\alpha.M=M \mbox{ and } S_\beta.M=M.$$

For simple root $$\alpha$$ (resp. $$\beta$$), we denote by $$M_\alpha$$ (resp. $$M_\beta$$) the Levi subgroup of its associated maximal parabolic subgroup. Denote by $$M'_\alpha$$ the derived group of $$M_\alpha$$, similarly $$M'_\beta$$.

The question about high-level Jacobson--Morozov SL_2 triple is as follows:

at least one of $$M'_\alpha$$ and $$M'_\beta$$ is isogenous to $$SL(n,D)$$?

where $$D$$ is a division algebra over $$F$$.

If $$G$$ is quasi-split and $$P$$ is a minimal parabolic subgroup, then it is isogenous to $$Res_{E/F}SL_2$$ or $$SU(2,1)_{F(\sqrt{d})/F}$$.

Thank you so much for your help.

• If you consider ${\mathbb R}$ as a local field, this is wrong for the Tits index $E_{6,2}^{28}$ (both Levi subgroups are of type $D_5$ then). For finite extensions of p-adic numbers your claim seems to be true by case-by-case considerations (not much possibilities for relative rank $2$ groups). – Victor Petrov Dec 5 '19 at 20:14
• Thanks! the relative root system is of type $A_1\times A_1$? If so, let us first exclude this situation. – chluo Dec 5 '19 at 20:49
• Jacobson-Morozov? – Dima Pasechnik Dec 5 '19 at 21:41
• Thanks! Corrected. – chluo Dec 5 '19 at 22:32
• No, the relative root system is $A_2$ in this case. – Victor Petrov Dec 6 '19 at 7:56