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Shapovalov elements $\theta _{\beta,m}$ of the classical or quantizeduniversal enveloping algebra of  a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root$\beta$ and a positive integer $m$. They relate the highest vector of areducible Verma module with highest vectors of its submodules. We obtain afactorization of $\theta_{\beta,m}$ to a product of $\theta_{\beta,1}$ andcalculate $\theta_{\beta,1}$ as a residue of a matrix element of the inverseShapovalov form via a generalized Nigel-Moshinsky algorithm. This way weexplicitly express $\theta_{\beta,m}$ of a classical simple Lie algebra throughthe Cartan-Weyl basis in $\mathfrak{g}$. In the case of quantum groups, we givean analogous formulation through the entries of the R-matrix (quantum$L$-operator) in fundamental representations.

Shapovalov elements of classical and quantum groups