The nephew David was considered illegitimate by the Georgian nobles. His father had seduced a married woman of non-noble status, brought her to court, and had a son by her in 1215. George was forced to return her to her husband (sending David to Rusudan to be raised), but refused any other suggestion of marriage by his councilors.
So far as the Georgian nobles could tell, the only available heir was the illegitimate David, since they knew where he was. He was also at that time 30 years old and able to understand his duties. He was crowned in 1245 as David VII, and known as David Ulu, "David the Senior."
A year later, Rusudan's son David returned from Karakoram with the blessing of Batu Khan. The 21-year-old became co-King with his cousin, given an earlier regnal number because of his precedence (son of the immediate previous ruler and born within wedlock) as King David VI Narin ("David the Junior").
The two managed to work together well until 1259. David VI rebelled independently against their Mongol overlords. It did not take many battles before he saw the error of his choices. Fleeing to western Georgia, where the Mongols did not have so much influence, he seceded from the rest of Georgia, being crowned in the western city of Kutaisi. (Kutaisi had been the capital of Georgia from c.1008 - 1122, when Tbilisi became the capital.)
Now, as King David I of Imereti (Western Georgia), he had control over several important ports on the Black Sea, as well as passes through the Caucasus Mountains, giving Western Georgia important trade routes.
Two years after David Narin's rebellion, David Ulu tried as well to become independent, joining his brother in Western Georgia. The two could not manage to agree on various policies, however, and David Ulu returned to Tbilisi in 1262, where he was forced to surrender to Hulagu Khan. Eastern Georgia was now a vassal, rather than an ally, of the Mongols.
Western Georgia was a different story, which will continue tomorrow.