The existence of an Iwasawa decomposition of split semisimple algebraic groups over the reals and the complex numbers is well known, as is the existence of an Iwasawa decomposition of split Kac-Moody groups over the reals and the complex numbers. There exist extensions of the theory of Iwasawa decompositions to real closed fields.

Using the theory of groups with a root group datum, the theory of Levi decompositions of parabolics of spherical type, and the theory of flips one can show that a split Kac-Moody group of two-spherical type over some field F admits an Iwasawa decomposition if and only if there exists a field automorphism σ of F of order one or two such that the sum of norms is a norm and -1 is not a norm with respect to the norm map F→ Fix_F(σ) sending x to x x^σ.