The notion of a reduced expression for a double coset in a Coxeter group was introduced by Williamson, and recent work of Elias and Ko has made this theory more accessible and combinatorial. One result of Elias-Ko is that any coset admits a reduced expression which factors through a reduced expression for a related coset called its core. In this paper we define a class of cosets called atomic cosets, and prove that every core coset admits a reduced expression as a composition of atomic cosets. This leads to an algorithmic construction of a reduced expression for any coset. In types $A$ and $B$ we prove that the combinatorics of compositions of atomic cosets matches the combinatorics of ordinary expressions in a smaller group. In other types the combinatorics is new, as explored in a sequel by Ko.