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 $\mathfrak{sp}{2n}(\mathbb {K})$.