Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R = k [(Formula presented)]. We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x1,…, xn) of R, and if this is the case, then the natural map (Formula presented)(R, G) (Formula presented)(k(x₁,..., x_n), G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that HZ_ar (R, G) = * for such groups G. We also deduce that if G is a reductive group over R of isotropic rank > 2, then the natural map of non-stable K₁-functors (Formula presented) ((x₁))...((x_n))) is injective, and an isomorphism if G is moreover semisimple.