Based on the second author's thesis [Horn, Involutions of Kac–Moody groups, TU Darmstadt, 2008], 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 in [Helminck, Wang, Adv. Math. 99: 26–96, 1993] and [Springer, Algebraic groups and related topics: 525–543, North-Holland, 1984] for algebraic groups and in [Kac, Wang, Adv. Math. 92: 129–195, 1992] for certain Kac–Moody groups, we analyze the filtration studied in [Devillers, Mühlherr, Forum Math. 19: 955–970, 2007] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [Bennett, Gramlich, Hoffman, Shpectorov, Curtis–Phan–Tits theory: 13–29, World Scientific, 2003].