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Let G be a real semisimple Lie group. The regular coadjoint orbits of G (acertain dense family of top-dimensional orbits) can be partitioned into afinite set of types. We show that on each regular orbit, the Iwasawadecomposition induces a left-invariant foliation which is isotropic withrespect to the Kirillov symplectic form. Moreover, the dimension of the leavesdepends only on the type of the orbit. When G is a split real form, thefoliations induced from the Iwasawa decomposition are actually Lagrangianfibrations with a global transverse Lagrangian section. For these orbits, weuse the structure of the Iwasawa decomposition to construct completelyintegrable systems.  In order to partition the orbits into types and construct isotropic nilpotentfoliations, we make a somewhat detailed study of the conjugacy classes ofCartan subalgebras, starting with the work of Sugiura, which may be ofindependent interest. In particular, we give a fairly explicit construction arepresentative of each conjugacy class.

Isotropic foliations of coadjoint orbits from the Iwasawa decomposition