  Let k be a field. Let G be an absolutely almost simple simply connected
k-group of type A_l, l>=2, or D_l, l>=4, containing a 2-dimensional split
torus. If G is of type D_l, assume moreover that char k is different from 2. We
show that the Nisnevich sheafification of the non-stable K_1-functor K_1^G,
also called the Whitehead group of G, on the category of smooth k-schemes is
A^1-invariant, and has oriented weak transfers for affine varieties in the
sense of Panin-Yagunov-Ross. If k has characteristic 0, this implies that the
Nisnevich sheafification of K_1^G is birationally invariant.
  We also prove a rigidity theorem for \A1-invariant torsion presheaves with
oriented weak transfers over infinite fields. As a corollary, we conclude that
K_1^G(R)=K_1^G(k) whenever R is a Henselian regular local ring with a
coefficient field k.
