Varieties of a class of elementary subalgebras

Let $G$ be a connected standard simple algebraic group of type $C$ or $D$ over an algebraically closed field $\mathbb{k}$ of positive characteristic $p>0$, and $\mathfrak{g}:=\mathrm{L}\mathrm{i}\mathrm{e}(G)$ be the Lie algebra of $G$. Motivated by the variety of $\mathbb{E}(r,\mathfrak{g})$ of $r$-dimensional elementary subalgebras of a restricted Lie algebra $\mathfrak{g}$, in this paper we characterize the irreducible components of $\mathbb{E}({\mathrm{r}\mathrm{k}}_p(\mathfrak{g})-1,\mathfrak{g})$ where the $p$-rank ${\mathrm{r}\mathrm{k}}_p(\mathfrak{g})$ is defined to be the maximal dimension of an elementary subalgebra of $\mathfrak{g}$.