Strong A 1 ${\mathbb {A}}^1$ -invariance of A 1 ${\mathbb {A}}^1$ -connected components of reductive algebraic groups

We show that the sheaf of $A^1$-connected components of a reductive algebraic group over a perfect field is strongly $A^1$-invariant. As a consequence, torsors under such groups give rise to $A^1$-fiber sequences. We also show that sections of $A^1$-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.