Complex sentence: In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).
Split into simpler sentences: In general, any proper rigid transformation can be decomposed as a rotation followed by a translation. Any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).