  We show that certain embeddings of Coxeter groups within other Coxeter groups
are injective using the notion of Coxeter partitions. Moreover, we study
Lusztig's partitions, which are generalizations of Lusztig's admissible maps
and Crisp's foldings. We show that they classify the simplest type of Coxeter
partitions, whose embeddings of Coxeter groups send each generator to a product
of commuting generators. Consequently, these embeddings are also injective, and
we prove that they preserve Coxeter numbers. These results were previously
known, due to work of M\"{u}hlherr and Dyer.
