The group SU₃ is Cayley

A linear algebraic group G is over a field K is called a Cayley 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 prove that the special unitary group in 3 variables SU_3 is a Cayley group over R, thus extending a result of V.L. Popov, who proved in 1975 that the special linear group in 3 variables SL_3 over an algebraically closed field of characteristic 0 is Cayley. We also discuss the question whether the R-group SL_{3,R} is Cayley.