  Based on the second author's thesis in this article we provide a uniform
treatment of abstract involutions of algebraic groups and of Kac-Moody groups
using twin buildings, RGD systems, and twisted involutions of Coxeter groups.
Notably we simultaneously generalize the double coset decompositions
established by Springer and by Helminck-Wang for algebraic groups and by
Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by
Devillers-Muhlherr in the context of arbitrary involutions, and we answer a
structural question on the combinatorics of involutions of twin buildings
raised by Bennett-Gramlich-Hoffman-Shpectorov.
