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A while back I was given a hardcopy of Casimir's thesis, and recently scanned it so that I could have an electronic version. The thesis doesn't appear to be readily available online, so thought it would be worth sharing here. You can download it from my public Dropbox here. The quadratic Casimir is introduced in Theorem III on page 93 of the thesis (page 52 of the scan) - the $\mathcal{D}_\mu$ are elements of the Lie algebra, considered as (right-invariant, I think) differential operators on $C^\infty(G)$, and $g^{\lambda\mu}$ is the inverse of the Killing metric with respect to this basis. Casimir actually considered the case of an arbitrary semi-simple Lie group, and not just the rotation group.

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A while back I was given a hardcopy of Casimir's thesis, and recently scanned it so that I could have an electronic version. The thesis doesn't appear to be readily available online, so thought it would be worth sharing here. You can download it from my public Dropbox here. The quadratic Casimir is introduced in Theorem III on page 93 of the thesis (page 52 of the scan) - the $\mathcal{D}_\mu$ are elements of the Lie algebra, considered as (right-invariant, I think) differential operators on $C^\infty(G)$, and $g^{\lambda\mu}$ is the inverse of the Killing metric with respect to this basis. Casimir actually considered the case of an arbitrary semi-simple Lie group, and not just the rotation group.