  We show that the sheaf of $\mathbb A^1$-connected components of a reductive
algebraic group over a perfect field is strongly $\mathbb A^1$-invariant. As a
consequence, torsors under such groups give rise to $\mathbb A^1$-fiber
sequences. We also show that sections of $\mathbb A^1$-connected components of
anisotropic, semisimple, simply connected algebraic groups over an arbitrary
field agree with their $R$-equivalence classes, thereby removing the
perfectness assumption in the previously known results about the
characterization of isotropy in terms of affine homotopy invariance of
Nisnevich locally trivial torsors.
