  Any irreducible Dynkin diagram $\Delta$ is obtained from an irreducible
Dynkin diagram $\Delta_h$ of type $\mathrm{ADE}$ by folding via graph
automorphisms. For any simple complex Lie group $G$ with Dynkin diagram
$\Delta$ and compact Riemann surface $\Sigma$, we give a Lie-theoretic
construction of families of quasi-projective Calabi-Yau threefolds together
with an action of graph automorphisms over the Hitchin base associated to the
pair $(\Sigma, G)$ . These give rise to Calabi-Yau orbifolds over the same
base. Their intermediate Jacobian fibration, constructed in terms of
equivariant cohomology, is isomorphic to the Hitchin system of the same type
away from singular fibers.
