  This article presents two key computations in MW-motivic cohomology. Firstly,
we compute the MW-motivic cohomology of the symplectic groups $Sp_{2n}$ for any
$n\in\mathbb{N}$ using the $Sp$-orientation and the associated Borel classes.
Secondly, following the classical computations and using the analogue in
$\mathbb{A}^1$-homotopy of the Leray spectral sequence, we compute the
$\eta$-inverted MW-motivic cohomology of general Stiefel varieties, obtaining
in particular the computation of the $\eta$-inverted MW-motivic cohomology of
the general linear groups $GL_n$ and the special linear groups $SL_n$ for any
$n\in\mathbb{N}$. Finally, we determine the multiplicative structures of these
total cohomology groups.
