ðə pɹˈuːf hˈæzbiːn fˈɜːðɚ sˈɪmplᵻfˌaɪd baɪ vˈɪktɚ kˈæk, sˌoʊ ðæt ˈoʊnli ðə kwɑːdɹˈæɾɪk kˈæsɪmˌɪɹ ˈɑːpɚɹˌeɪɾɚɹ ɪz ɹᵻkwˈaɪɚd; ðɛɹ ɪz ɐ kˌɔːɹɪspˈɑːndɪŋ stɹˈiːmlaɪnd tɹˈiːtmənt pɹˈuːf ʌvðə kˈæɹɪktɚ fˈɔːɹmjʊlə ɪnðə sˈɛkənd ᵻdˈɪʃən ʌv .
The proof has been further simplified by Victor Kac, so that only the quadratic Casimir operator is required; there is a corresponding streamlined treatment proof of the character formula in the second edition of .