Non-stable K₁-functors of multiloop groups

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}] containing a maximal R-torus T (equivalently, loop reductive). Assume also that every semisimple normal subgroup of G contains a two-dimensional split torus G_m^2. We show that the natural map of non-stable K_1-functors K_1^G(R)-> K_1^G(k((x_1))...((x_n))) is injective. This complements the surjectivity result for the same map obtained by V. Chernousov, P. Gille and A. Pianzola in arXiv:1109.5236. As a corollary, we provide a way to evaluate the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.