Some new results concerning the symplectic and algebraic geometry of nilpotent orbits of a complex semisimple Lie group are presented. Let \(G\) be a simply connected complex semisimple Lie group and \({\mathfrak g}\) the Lie algebra of \(G\). Let \(e\in{\mathfrak g}\) be nilpotent and let \(O\) be the adjoint orbit of \(e\) and \(\nu: M\to O\) a \(G\)-covering. The ring \(R=R(M)\) of regular functions on \(M\) carries a \(G\)-invariant grading \(R=\oplus_{k\geq 0}R_ k\), \(k\in\mathbb{Z}\). If \(R[{\mathfrak g}]\) is the copy of \({\mathfrak g}\) in \(R\), then \(R[{\mathfrak g}]\subset R[2]\). It is proven that \({\mathfrak g}'=R[2]\) is a semisimple Lie algebra and is simple if \({\mathfrak g}\) is simple. The complete list of all pairs \(({\mathfrak g},{\mathfrak g}')\) for all simple \({\mathfrak g}\)’s is given. Some further results concerning the structure of the ring \(R\) are presented. An application to the symmetry problem of flag varieties is given.