  A linear algebraic group G over a field k is called a Cayley group if it
admits a Cayley map, i.e., a G-equivariant birational isomorphism over k
between the group variety G and the Lie algebra Lie(G). A Cayley map can be
thought of as a partial algebraic analogue of the exponential map. A
prototypical example is the classical "Cayley transform" for the special
orthogonal group SO_n defined by Arthur Cayley in 1846. A k-group G is called
stably Cayley if the product of G with a split r-dimensional k-torus is Cayley
for some r=0,1,2,.... These notions were introduced in 2006 by N. Lemire, V. L.
Popov and Z. Reichstein, who classified Cayley and stably Cayley simple groups
over an algebraically closed field of characteristic zero.
  In this paper we study Cayley and stably Cayley reductive groups over an
arbitrary field k of characteristic zero. Our main results are a criterion for
a reductive k-group G to be stably Cayley, formulated in terms of its character
lattice, and the classification of stably Cayley simple (but not necessarily
absolutely simple) groups.
