  The Cartan and Iwasawa decompositions of real reductive Lie groups play a
fundamental role in the representation theory of the groups and their
corresponding symmetric spaces. These decompositions are defined by an
involution with a compact fixed-point group, called a Cartan involution. For an
arbitrary involution, one can consider similar decompositions. We offer a
generalization of the Cartan and Iwasawa decompositions for algebraic groups
defined over an arbitrary field $k$ and a general involution.
