On nilpotent generators of the symplectic Lie algebra

Abstract

Let $\\mathfrak{sp}_{2n}(\\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X \\in \\mathfrak{sp}_{2n}(\\mathbb {K})$ there exists a nilpotent element $Y \\in \\mathfrak{sp}_{2n}(\\mathbb {K})$ such that $X$ and $Y$ generate $\\mathfrak{sp}_{2n}(\\mathbb {K})$.