I believe it was commonly accepted at the time that the Imperial title had to keep both Aachen and Rome under them, as the two historic capitals of Francia and the Roman Empires. That is why Lotharingia was such a crappy shape in the first place.
Middle Francia (Lotharingia is only the part North of Burgundy) was shaped in such a way that it gave all three brothers a share of the core Frankish Lands, Austrasia and Neustria. Louis the German already was (sub-) king of Bavaria and Lothar was (sub-)king of Italy, co-emperor and after the death of Louis the Pious sole emperor. Louis the Pious had promised his youngest son, Charles, half-brother of Lothar and Louis the German, the western part of the realm.
@Intosh: I partially answered to
@Skallagrim suggestion to have an ATL two way division, there Aquitania will very much belong to the Western half. The border between East and West, that can remain open for debate. It can be anywhere between the West only getting Neustria and the East ending up with the whole of Austrasia, to your suggestion of a Rhine border, or something like the situation after the OTL treaty of Meerssen, a Meuse/Maas border (so somewhere in the middle of the previous two suggestions.
That can depend on a few things, which son is the favourite of the previous ruler, or which one ended up rebelling. If the eldest gets a Rhine border, then he was the favourite or the youngest overplayed his hand. If OTOH the youngest gets the entirety of Austrasia (the blue line division in this thread), then the youngest was the favourite or the eldest overplayed his hand.
This is all assuming an ATL two way split. A threeway split like IOTL could easily have gone differently, a more successful Lothar against Louis the German and Charles the Bald would have been able to increase his share at the expense of his younger brothers. Under such a scenario Louis would have gotten less of Austrasia (Rhine border perhaps) and Charles could only have gotten Neustria, which would under that scenario increase the share of the now victorious Lothar. While after that Middle Francia could still fracture under such a scenario.