  Let G be a noncompact real semisimple Lie group. The regular coadjoint orbits
of G can be partitioned into a finite set of types. We show that on each
regular orbit, the Iwasawa decomposition induces a left-invariant foliation
which is isotropic with respect to the Kirillov symplectic form. Moreover, the
leaves are affine subspaces of the dual of the Lie algebra, and the dimension
of the leaves depends only on the type of the orbit. When G is a split real
form, the foliations induced from the Iwasawa decomposition are actually
Lagrangian fibrations with a global transverse Lagrangian section.
