  Let $G$ be a reductive algebraic group over a perfect field $k$ and $\cG$ a
$G$-bundle over a scheme $X/k$. The main aim of this article is to study the
motive associated with $\cG$, inside the Veovodsky Motivic categories. We
consider the case that $\charakt k=0$ (resp. $\charakt k\geq 0$), the motive
associated to $X$ is geometrically mixed Tate (resp. geometrically cellular)
and $\cG$ is locally trivial for the Zariski (resp. \'etale) topology on $X$
and show that the motive of $\cG$ is geometrically mixed Tate. Moreover for a
general $X$ we construct a nested filtration on the motive associated to $\cG$
in terms of weight polytopes. Along the way we give some applications and
examples.
