Why, out of the three parts of Charlemagne’s empire, it was Germany (and not France or Italy) that became the center

Why, out of the three parts of Charlemagne’s empire, it was Germany (and not France or Italy) that became the center of the revival of the empire in the X century?

The center of Karl’s power was the future Northern France (then Austrasia) – a country united by Clovis and earlier than others who formalized its statehood. After Charles’ death, the French eagerly separated from the newly born empire they did not need. On the contrary, the Germans began to build their statehood precisely within the framework of the empire, they considered it their own.