A new approach to computing generators for UK(g)

Let (G,K) be the complex symmetric pair associated with a real reductive Lie group G0. We discuss an algorithmic approach to computing generators for the centralizer of K in the universal enveloping algebra of g. In particular, we compute explicit generators for the cases G0=SU(2,2), SL(3,R), SL(4,R), Sp(4,R), and the exceptional group G2(2).