Let \(G\) be a simple algebraic group over an algebraically closed field \(\mathfrak k\) of characteristic \(0\) and let \({\mathcal O} = G\cdot x\), \(x\in\mathfrak g\), be a nilpotent orbit. In [D. Panyushev, Manuscr. Math. 83, 223–237 (1994; Zbl 0822.14024)], the author showed \({\mathcal O}\) is \(G\)-spherical if and only if the orbit has height \(2\) or \(3\). In the case of height \(3\), the author’s proof depended on the classification of orbits. The author now provides a general conceptual argument. The author does this by examining the invariants of symplectic representations and applying the results to give the structure of \({\mathfrak z}_{\mathfrak g}(x)\) when the height is odd. A number of other results are also included. For instance, it is shown that \(\text{ind}({\mathfrak z}_{\mathfrak g}(x))=\text{rk}({\mathfrak g})\) when the height is \(3\) which confirms Elashvili’s conjecture for such \(x\). Moreover if the highest root is fundamental, a distinguished orbit of height \(3\) is studied. Finally, a number of results and some discussion is given to the problem of computing the algebra of covariants on a nilpotent orbit including a conjecture that will solve the problem in the case of height \(3\).