  Completely reducible subcomplexes of spherical buildings was defined by J.P.
Serre and are used in studying subgroups of reductive algebraic groups. We
begin the study of completely reducible subcomplexes of twin buildings and how
they may be used to study subgroups of algebraic groups over a ring of Laurent
polynomials and Kac-Moody groups by looking at the Euclidean twin building
case.
