On nilpotent generators of the special linear Lie algebra

Abstract

Consider the special linear Lie algebra $\mathfrak{sl}_n(\mathbb {K})$ over an infinite field of characteristic different from $2$. We prove that for any nonzero nilpotent $X$ there exists a nilpotent $Y$ such that the matrices $X$ and $Y$ generate the Lie algebra $\mathfrak{sl}_n(\mathbb {K})$.