Wikipedia says,
"If $G$ is a Lie group with Lie algebra $\mathfrak {g}$, the choice of an invariant bilinear form on $\mathfrak {g}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. Then under the identification of the universal enveloping algebra of $\mathfrak {g}$ with the left invariant differential operators on $G$, the Casimir element of the bilinear form on $\mathfrak {g}$ maps to the Laplacian of $G$ (with respect to the corresponding bi-invariant metric)."
Actually, a stronger claim is true for homogeneous spaces (and not only for the Lie group itself).
Do you have a reference to this claim, or even better, to a version of a stronger claim regarding homogeneous spaces?
You might find these notes by Paul Garrett helpful.