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@MartinGisser: Thanks, I think. Interestingly, right about the time you posted your comment (which I took to be positive, but maybe I misunderstood your intent), someone downvoted my answer! Go figure.
@AliTaghavi: The original references are, first, É Cartan, Les sous-groupes de groupes continus de transformations, Annales scientifiques de l'É.N.S. 3 série, t. 25 (1908), p.57–194., in which Cartan refers to prior works of Lie (Nouv. Arch. t. III, 1878, p. 125–; Nouv. Arch. t. X, 1885, p. 74–; Math. Ann. t. XVI, 1888, p. 455– ). See the list at the end of Cartan's paper, but one has to sort through which are transitive, primitive, and finis. Cartan and Lie assumed everything was analytic and all variables were complex, though — defects that the more modern authors were able to remedy.
@YCor: To modern standards of rigor, it's probably due to a combination of Goldschmidt, Olver, Fels, and/or Kamran, though that's based on Guillemin and Sternberg's reworking of Cartan's work on Lie transformation groups, itself building on Lie's original work. Certainly, if I wanted to give a short proof, I would need to refer to them, but I'd have to search to find the best literature reference. However, you don't really need that whole apparatus to prove the above statement. One can prove it by hand without too much difficulty. If there's interest I could sketch the argument.
@DavidSchwein: If you are interested in the history, there is a very nice 2008 article by Ilka Agricola about the history of the 'discovery' of $\mathrm{G}_2$ and its properties , ams.org/notices/200808/tx080800922p.pdf.