On nilpotent generators of the special linear Lie algebra

Consider the special linear Lie algebra ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{K})$ 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 ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{K})$.