Cayley groups

In 1886, Cayley discovered that the formula $X \dashrightarrow (I-X)/(I+X)$ defines a birational isomorphism between the special orthogonal group $SO(n)$ and its Lie algebra that is equivariant with respect to the adjoint actions. An algebraic group is called Cayley group if it admits a map with these properties. We consider the problem of classifying Cayley groups. As an application, one obtains some examples ofisomorphic but nonconjugate algebraic subgroups of Cremona group, in particular finite such subgroups.