I'm surprised that no one has mentioned the differential-geometric motivation for the Casimir element.

Suppose $\mathfrak{g}$ is the Lie algebra of a compact Lie group $G$. Then $U(\mathfrak{g})$ is the algebra of left-invariant differential operators on $G$.

$G$ has a natural bi-invariant metric--given by the Killing form. But now there is an obvious central element of $U(\mathfrak{g})$: the Laplacian associated to this metric! This is precisely the Casimir element; I had always assumed this was the original motivation for the construction.

I'm surprised that no one has mentioned the differential-geometric motivation for the Casimir element.

Suppose $\mathfrak{g}$ is the Lie algebra of a simple compact Lie group $G$. Then $U(\mathfrak{g})$ is the algebra of left-invariant differential operators on $G$.

$G$ has a natural bi-invariant metric, given by the Killing form. But now there is an obvious central element of $U(\mathfrak{g})$: the Laplacian associated to this metric! This is precisely the Casimir element. It's centrality follows "by pure thought" from the bi-invariance of the Killing form. To obtain Casimir elements in general and prove their centrality, simply copy the formulas from the case of the Lie algebra associated to a Lie group.

I had always assumed this was the original motivation for the construction.