Real reductive Cayley groups of rank 1 and 2

A linear algebraic group G over a field K is called a Cayley K-group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups of absolute rank 1 and 2 that are Cayley R-groups.