Split Milnor-Witt Motives and its Applications to Fiber Bundles

We study the Milnor-Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/\eta(q)[p]$. We show that for MW-motives of this type, we could determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroup of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real complete flags has only 2-torsions.