Motives of reductive groups

Our aim is to prove that the geometric motive with rational coefficients of a split reductive group $G$ over a perfect field decomposes as the symmetric co-algebra object of certain split mixed Tate motives which are computable in terms of the root data of $G$. Moreover the isomorphism respects the Hopf algebra structures in a weight graded sense. We will also provide explicit decompositions in the case of classical groups.