  Let $G$ be a simple algebraic group of type $F_{4}$, $E_{6}$, $E_{7}$ or
$E_{8}$, and let $\mathfrak{g}$ be its Lie algebra. The adjoint variety $X_{ad}
\subseteq \mathbb{P} \mathfrak{g}$ is defined as the unique closed orbit of the
adjoint action of $G$ on $\mathbb{P}\mathfrak{g}$. $X_{ad}$ is a Fano contact
manifold covered by lines in $\mathbb{P} \mathfrak{g}$. The subadjoint variety
$S \subseteq \mathbb{P} W$ is denoted by the variety of lines on $X_{ad}$
through a fixed point $x$, where $W \subseteq T_{x}X$ is taken as the contact
hyperplane. It follows from a result in representation theory of Vinberg that
the GIT quotient space of codimension two linear sections of $S$ is isomorphic
to the weighted projective space $\mathbb{P}(1,3,4,6)$. In this note, we
investigate the problem of finding a geometric interpretation of the above
isomorphism. As a main result, for each $\mathfrak{g}$ of the above type, we
construct a natural open embedding of the GIT quotient space of nonsingular
codimension two linear sections of $S$ into $\mathbb{P}(1,3,4,6)$ whose
complement is a fixed hypersurface of degree 12. The key ingredient of our
construction is to apply a correspondence of Bahargava and Ho which relates the
above moduli problem to a moduli problem on curves of genus one.
