Stably Cayley semisimple groups

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 its Lie algebra Lie(G). A prototypical example is the classical "Cayley transform" for the special orthogonal group \boldSOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if G×S is Cayley for some split k-torus S. We classify stably Cayley semisimple groups over an arbitrary field k of characteristic 0.