Under what treaty did the Isle of Man become part of Scotland?

The Treaty of Perth, signed 2 July 1266, ended military conflict between Magnus VI of Norway and Alexander III of Scotland over the sovereignty of the Hebrides and the Isle of Man.

The Hebrides and the Isle of Man had become Norwegian territory during centuries when both Scotland and Norway were still forming themselves as coherent nation-states, and Norwegian control had been formalised in 1098, when Edgar of Scotland signed the islands over to Magnus III of Norway. In Norwegian terms, the islands were the Sudreys, meaning Southern Isles.

The Treaty was agreed three years after the Battle of Largs in 1263. Michael Lynch has compared the treaty's importance with that of the Treaty of York of 1237. The Treaty of York defined a border between Scotland and England which is almost identical to the modern border.