  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.
