Complex sentence: Lübeck remained part of Schleswig-Holstein after the war (and consequently lay within West Germany) and was situated directly on the inner German border during the division of Germany into two states in the Cold War period.
Split into simpler sentences: Lübeck remained part of Schleswig-Holstein after the war (and consequently lay within West Germany). It was situated directly on what became the inner German border during the division of Germany into two states in the Cold War period.