  Using weak solutions to the conjugation equation, we define a fibered
rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d}
\times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove
that if this rotation vector is Diophantine with respect to the rotation in
$\mathbb{T}^{d}$, the cocycle is smoothly reducible, thus establishing a
hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in
PDEs.
