What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

Apologies if it isn't customary to answer questions asked and answered so long ago, I'm new here...

In his autobiography Het toeval van de werkelijkheid (1983), Casimir makes a short remark about his operator (there is an English translation, Haphazard reality, but I don't own it so this translation is my own):

My dissertation [...] also describes wat is nowadays often called the Casimir operator. To put this in the proper perspective, I would like to quote something from a witty English book called How to be famous. Concerning Plato, it says: 'His own inaccurate ideas about platonic love are outdated. In his own time, however, they were an improvement on the existing situation, where one had no concept of platonic friendship whatsoever.' Mutatis mutandis, the same could be said about Casimir operators.

This at least tells us he thought he was the first to find the Casimir operator, but he is very modest about his role in the general theory; he doesn't seem to have worked on it besides the case of the rotation group, and I think he says somewhere else that he feels it isn't fair that the operator is named after him.

So I'd agree (and so would Casimir himself): he was the first to find an example, but wasn't involved in the subsequent development.

At the first glance it appears that he more or less just gave the first nontrivial example(s) of what was later called the Casimir operators.

His obituary says:

On 1 May 1931 he wrote a letter from The Hague to the famous Gottingen mathematician Hermann Weyl and announced: ‘While studying the quantum-mechanical properties of the asymmetic rotator I arrived at some ‘results’ (?) concerning the representation of continuous groups.’ He then sketched his findings on the matrix elements of the irreducible representations for the three-dimensional rotation group, and a possible extension for semi-simple groups in general, where he introduced what was later called the ‘Casimir operator’. This operator turned out to be a multiple of the unit-operator and may be used to characterize in an elegant way the irreducible representations of a given continuous group. To Casimir’s question, ‘Whether the case is worth considering?’, Weyl answered definitely ‘Yes’. Hence the Leiden doctoral candidate published his mathematical results in a paper, communicated by Ehrenfest to the meeting of 27 June 1931 of the Amsterdam Academy [7], and he also included them as Chapter IV of his dissertation, which he defended on 2 November 1931 at the University of Leiden [8].

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.

EDIT: adding a Scribd link here for those who would prefer not to download the entire 25 meg file.