De primo Saxonum adventu claims that Oslac, along with Eadulf of Bamburgh and Ælfsige Bishop of Chester-le-Street, escorted the Scottish king Kenneth II to the Wessex-based Edgar: The two earls Oslac and Eadwulf along with Ælfsige, who was bishop of St Cuthbert 968—90, conducted Cinaed to king Edgar.
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On the other hand Oslac is also a genuine English name, and the common Os element Oslac's name shared with the name of Osulf of Bamburgh, previous ealdorman of York, points to a connection with the Bamburgh family of the English far north.
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De primo Saxonum adventu, an 11th- or 12th-century compilation from earlier sources, notes that after the death of Osulf, Northumbria was divided into two parts: Eadulf Evil-child receiving the lands between the Myreford (arguably the Firth of Forth) and the River Tees and Oslac receiving the lands between the Humber Estuary and the Tees.