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For $G$ a connected, reductive group over an algebraically closed field $k$of large characteristic, we use the canonical Springer isomorphism between thenilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent varietyof $G$ to study the projective variety of elementary subalgebras of rank $r$,denoted $\mathbb{E}(r,\mathfrak{g})$. In the case that $G$ is defined over$\mathbb{F}_p$, we define the category of $\mathbb{F}_p$-expressiblesubalgebras of $\mathfrak{g}$, and prove that this category is isomorphic toQuillen's category of elementary abelian subgroups of the finite Chevalleygroup $G(\mathbb{F}_p)$. This isomorphism of categories leads to acorrespondence between $G$-orbits of $\mathbb{E}(r,\mathfrak{g})$ defined over$\mathbb{F}_p$ and $G$-conjugacy classes of elementary abelian subgroups ofrank $r$ in $G(\mathbb{F}_p)$. We use Magma to compute examples for$G=\mathrm{GL}_n$, $n\le 5$.

$\mathbb{F}_p$-expressible subalgebras and orbits of $\mathbb{E}(r,\mathfrak{g})$