I will toot my own horn a bit by mentioning the lecture notes for a summer course I taught on this very subject last year.  The relevant lecture is [this one][1] and in it, I decided (based on answers given to my question [Why the Killing form?][2]) to present Casimir as an analogue of the averaging operation for proving "Weyl's theorem" (i.e. Maschke's theorem) for finite groups.  I never did find a good way of actually drawing an exact parallel, but I did try to structure the proof in such a way that it was clear this was how it functioned.  (All the actual math is copied from Humphreys.)


  [1]: http://math.harvard.edu/~ryanr/algebraic_groups/2010-07-25(lecture-1).pdf
  [2]: https://mathoverflow.net/questions/32554/why-the-killing-form