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We prove a global Birkhoff decomposition for almost split real forms of loopgroups, when an underlying finite dimensional Lie group is compact. Amongapplications, this shows that the dressing action - by the whole subgroup ofloops which extend holomorphically to the exterior disc - on the $U$-hierarchyof the ZS-AKNS systems, on curved flats and on various other integrablesystems, is global for compact cases. It also implies a global infinitedimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1wave maps) from surfaces into compact symmetric spaces. An "Iwasawa-type"decomposition of the same type of real form, with respect to a fixed pointsubgroup of an involution of the second kind, is also proved, and anapplication given.

Loop Group Decompositions in Almost Split Real Forms and Applications to Soliton Theory and Geometry