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In earlier work, Barchini, Kable, and Zierau constructed a number ofconformally invariant systems of differential operators associated toHeisenberg parabolic subalgebras in simple Lie algebras. The construction wassystematic, but the existence of such a system was left open in two cases,namely, the $\Omega_3$ system for type $A_2$ and type $D_4$. Here, such asystem is shown to exist for both cases. The construction of the system mayalso be interpreted as giving an explicit homomorphism between generalizedVerma modules.

A System of Third-Order Differential Operators Conformally Invariant under $\mathfrak{sl}(3,\mathbb{C})$ and $\mathfrak{so}(8,\mathbb{C})$