It is well known that Verma module homomorphisms correspond to invariant operators on homogeneous spaces, which in certain situations can be regarded as the flat models of specific differential geometries. This can be generalised to curved space by introducing semiholonomic Verma modules, whose homomorphisms give rise to invariant operators on curved space. In this article we investigate from a purely algebraic point of view which Verma module homomorphisms lift to the semiholonomic case for the exceptional Lie algebra $E_6$.