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In the 'Treaty of Verdun' of 843 the three surviving sons of Louis the Pious, Charles the Great's (Charlemagne's) grandsons, divided his territories, the Carolingian Empire, into three kingdoms. Though often presented as the beginning of a devolution or dissolution of Charlemagne's unitary empire, it in fact reflected the continued adherence to the Germanic, and therefore Frankish, idea of a partible or divisible inheritance rather than primogeniture, inheritance by the eldest son.
When Louis the Pious died in 840, the eldest son, Lothair I, claimed overlordship over his brothers' kingdoms and supported the claim of his nephew Pepin II as king of Aquitaine. After his brothers Louis the German and Charles the Bald defeated his forces at the Battle of Fontenay (841) and sealed their alliance with the Oath of Strasbourg (842), Lothair was willing to negotiate.
Each of the brothers was already established in one kingdom - Lothair in Italy, Louis the German in Bavaria, and Charles the Bald in Aquitaine.

★ Lothair received the central portion of the empire - what later became the Low Countries, Lorraine, Alsace, Burgundy, Provence, and Italy, and the imperial title as an honour without more than nominal overlordship. Lothair received the Imperial title, the Kingship of Italy, and the territory between the Rhine and Rhône Rivers, collectively called the Middle Frankish Kingdom.

★ Louis the German received the eastern portion, much of what later became today's Germany through the intermediate collection of states, the Holy Roman Empire. Louis was guaranteed the kingship of all lands to the east of the Rhine and to the north and east of Italy, which was called the Eastern Frankish Realm which was the precursor to the Medieval conglomeration of disparate states known as the Holy Roman Empire and thence to modern Germany.

★ Charles the Bald received the western portion, which later became France. Pepin II was granted the kingdom of Aquitaine, but only under the authority of Charles. Charles received all lands west of the Rhône, which was called the Western Frankish Realm.
Lothar retired Italy to his eldest son Louis II in 844, making him co-Emperor in 850. Lothar died in 855, dividing his kingdom into three parts: the territory already held by Louis remained his, the territory of the former Kingdom of Burgundy was granted to his third son Charles of Provence, and the remaining territory to his second son Lothar II, after whom the hitherto nameless territory was called Lotharingia.
Louis II, dissatisfied with having received no additional territory with his father's death, allied with his uncle Louis the German against his brother Lothar and his uncle Charles the Bald in 858. Lothar was reconciled with his brother and uncle shortly after, though Charles was so unpopular he could not raise an army to fight the invasion and fled to Burgundy, he was only saved when the bishops refused to crown Louis the German King. Charles the Bald invaded Charles of Provence's Kingdom of Burgundy in 860, but was repulsed. Lothar II ceded lands to Louis II in 862 for support of a divorce from his wife, which caused repeated conflicts with the Pope and his uncles. Charles of Burgundy died in 863, and his Kingdom was inherited by Louis II.
Lothar II died in 869 with no legitimate heirs, and his Kingdom was divided between Charles the Bald and Louis the German in 870 by the Treaty of Meerssen. Meanwhile, Louis the German was involved with disputes with his three sons. Louis II died in 875, and named Carloman, the eldest son of Louis the German, his heir. Charles the Bald, supported by the Pope, was crowned both King of Italy and Holy Roman Emperor. The following year, Louis the German died. Charles tried to annex his realm too, but was defeated decisively at Andernach, and the Kingdom of the eastern Franks was divided between Louis the Younger, Carloman of Bavaria and Charles the Fat.