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 and in it, I decided (based on answers given to my question Why the Killing form?) 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.)

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 and in it, I decided (based on answers given to my question Why the Killing form?) 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.)