The paper focuses on an explicit determination of a set of generators for polynomial invariants. Generalized Casimir operators are introduced for the dual pairs \((\text{SO}^*(2n),\text{Sp}(2k,\mathbb{C}))\) and \((\text{Sp}(2n,\mathbb{R}),O(N))\), for generating the algebra of all \(\mathcal G\) invariant differential operators of the universal enveloping algebra \({\mathcal U}({\mathcal H}^{(1)} \times \dots \times {\mathcal H}^{(m)})\), \({\mathcal H}^{(s)} = \text{gl}(2k,\mathbb{C})\) being the Lie algebras \(1 \leq s \leq m\). Let \(G = \text{Sp}(2k,\mathbb{C})\) or \(O(N)\) and \(G' = SO^*(2n)\) or \(\text{Sp}(2n,\mathbb{R})\), then the adjoint representation of \(G'\) on its Lie algebra \({\mathcal G}'\) gives rise to the coadjoint representation on the symmetric algebra of all polynomial functions on \({\mathcal G}'\). The algebras of all \(K'\)-invariant polynomials of \(G' = \text{SO}^*(2n)\) or \(\text{Sp}(2n,\mathbb{R})\) are constructed for a block diagonal subgroup of \(G'\). Using the theory of invariants of Procesi for the “dual pair” \((G',G)\), a finite set of generators of this algebra is explicitly determined.