At the first glance it appears that he more or less just gave the first nontrivial example(s) of waht what was latter later called the Casimir operators.
His obituary says:
On 1 May 1931 he wrote a letter from The Hague to the famous G¨olttingen 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].
