A¹-connectedness in reductive algebraic groups

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Using sheaves of A¹-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A¹-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of A¹-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize A¹-connected reductive algebraic groups over a field of characteristic 0.