  Rational loops played a central role in Uhlenbeck's construction of harmonic
maps into U(n) (chiral model in physics), and they are generated by simple
elements with one pole and one zero constructed from Hermitian projections. It
has been believed for long time that nilpotent loops should be added to
generate rational loop groups with noncompact reality conditions. We prove a
somewhat unexpected theorem that projective loops are enough to generate the
rational loop groups of GL(n,C), GL(n,R), and U(p, q).
