3 added 28 characters in body

I think ${\rm Aut}(\Delta(B,T))$ doesn't always split: there are some very serious obstructions for that.

According to a recent result of Skip Garibaldi ad Philippe Gille on algebraic groups with few subgroups, published in PLMS, there exist groups $G$ of type ${\rm D}_4$ whose all proper connected subgroups defined over the base field are abelian (such groups are necessarily anisotrpic). If the extension in question splits then $\rm Aut(G)$ contains an involution which acts on the Dynkin graph by interchanging two nodes and fixing the rest. The fixed point group of such an involution would contain a connected reductive subgroup or semisimple rank $3$ defined over the base field. However, such subgroups cannot exist in the forms of type ${\rm D}_4$ constructed by Garibaldi and Gille.

2 added 3 characters in body

I think ${\rm Aut}(\Delta(B,T))$ doesn't always split: there are some very serious obstructions for that.

According to a recent result of Skip Garibaldi ad Philippe Gille on algebraic groups with few subgroups, published in PLMS, there exist groups $G$ of type ${\rm D}_4$ whose all proper connected subgroups are abelian (such groups are necessarily anisotrpic). If the extension in question splits then $\rm Aut(G)$ contains an involution which acts on the Dynkin graph by interchanging two nodes and fixing the rest. The fixed point group of such an involution would contain a connected reductive subgroup or semisimple rank $3$ defined over the base field. However, this is is not the case for such subgroups cannot exist in the forms of type ${\rm D}_4$ constructed by Garibaldi and Gille.

1

I think ${\rm Aut}(\Delta(B,T))$ doesn't always split: there are some very serious obstructions for that.

According to a recent result of Skip Garibaldi ad Philippe Gille on algebraic groups with few subgroups, published in PLMS, there exist groups $G$ of type ${\rm D}_4$ whose all proper connected subgroups are abelian (such groups are necessarily anisotrpic). If the extension in question splits then $\rm Aut(G)$ contains an involution which acts on the Dynkin graph by interchanging two nodes and fixing the rest. The fixed point group of such an involution would contain a connected reductive subgroup or semisimple rank $3$ defined over the base field. However, this is is not the case for the forms of type ${\rm D}_4$ constructed by Garibaldi and Gille.