From what I've read, American Bantam cobbled the first Jeep prototype together. They didn't have capacity to make the number required. The gov't liked the idea, took it to Willys-Overland and Ford to see how it might be developed for mass production. The Willys-Overland revised prototype was selected, with some Ford modifications (like the iconic grille, for example). Willys-Overland wasn't able to build enough for the government, either, as it turned out so Ford was licensed to build the Willys final design. And made more of them during the war than Willys. Bantam was given a contract to make cargo trailers. I've never read that GM had any role in Jeep development or production.
Oh, and military Jeeps normally used OE 30 engine oil for lubricant.
