Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For example a motivic decomposition of $F\_4/P$, with $P$ being a certain parabolic subgroup has been established by Semenov, Nikolenko and Zainoulline.


Looking at the Dynkin diagramm of $D\_n$ one will notice that the case $n=4$ gives a diagramm with a special symmetry (cue: triality). 


Question: Is the symmetry of the Dynkin diagramm of $D\_4$ somehow reflected in the motive of $D\_4$ or is there a motivic decomposition known at all?


It might be possible that the motivic structure can be derived from general results, but i dont too many papers dealing with these kinds of problems. 



Motives of a variety of type D4
You can compute the motive of $SO(8)$ with rational coefficients in the category of derived mixed Tate motives with Biglari's theorem (after looking up the exponents of the Weyl group in Bourbaki's tables).


The exponents are $1,3,5,3$, which gives the numbers $2,4,6,4$ as dimensions of certain cohomology groups of weight graded parts of $M(SO(8))$.


The primitive motive of $SO(8)$ therefore is
$$PM(SO(8)) = \mathbb{Q}(2)[3] \oplus \mathbb{Q}(4)[7] \oplus \mathbb{Q}(4)[7] \oplus \mathbb{Q}(6)[11]$$
and then we have
$$M(SO(8))\_{\mathbb{Q}} = Sym^\bullet PM(SO(8)).$$


I don't immediately see the triality operation in this description, but it should be there. Maybe you can get a better hold on triality by looking at $M(SO(8)/B)$ in terms of the Bruhat decomposition...


added in response to Matthias' comment:
In Gross' article, he gives an ad-hoc definition for "the motive of $G$", which can be compared with the primitive motive of $G$ by the expression
$$M^{Gross}(G)(1) = PM(G)$$
Also note that Gross uses another index convention.


