Let $$G=SO(n,1):=\{A\in\text{Mat}_{n+1,n+1}(\Bbb R)\colon\langle Av,Aw\rangle=\langle v,w\rangle\ \forall v,w\in\Bbb R^{n+1}\}$$ where $$\langle\sum_{i=1}^{n+1}\lambda_ie_i, \sum_{j=1}^{n+1}\mu_je_j\rangle:=\sum_{i=1}^n\lambda_i\mu_i-\lambda_{n+1}\mu_{n+1}$$ with the standard basis $(e_i)_{i=1}^{n+1}$ of $\Bbb R^{n+1}$. Let $\mathfrak g$ be the corresponding Lie algebra. I wonder if the center of the universal enveloping algebra $U(\mathfrak g)$ is generated by the quadratic Casimir operator.
I know that if $K:=S(O(n)\times O(1))\cong O(n)\leq G$ denotes the maximal compact subgroup, the set of invariant differential operators $D(G/K)$ on $G/K$ consists of polynomials in the Laplace-Beltrami operator (see [Helgason, Groups and geometric analysis, p.288, Prop. 4.11]) which comes from the quadratic Casimir element.