Abstract
Let \(V\) be the 56-dimensional minuscule representation of \(G=E_7\), with Lie algebra \(\mathfrak{g}\), and put \(M(x)=\rho(x)\). Trace-projected multiplication by \(M\), from skew-adjoint covariants to trace-free self-adjoint covariants, is surjective and has a free rank-one kernel over \(A=\mathbb{C}[\mathfrak{g}]^G\), generated by \(K_{27}=\text{adj}(M)/P_{28}\), where \(P_{28}^2=\det M\). The generator has polynomial degree 27, corresponding to degree 29 in the tensor-placement grading. A relative-invariant construction gives the kernel for every nontrivial irreducible self-dual minuscule representation. Within this class, we classify all simple and semisimple exact cases, both symplectic and orthogonal, by a root-reflection criterion. For the chosen half-product, the kernel class on the Freudenthal variety is \(\overline{K}_{27}=-2[\text{pt}]=\overline{M}^{27}/6555\); the exact orthogonal cases give terminal classes in Lefschetz strings. In the self-dual minuscule setting, the cokernel is the normalization quotient of the weight divisor in the adjoint quotient. We give explicit prime conductor ideals for the first three nonexact spin cases and a uniform conductor formula proving reducedness of the annihilator-defined support. Independently of minusculity, we prove that invariant-gradient operators generate, over the invariant ring, the center of the classical family algebra for every irreducible finite-dimensional representation of a connected complex reductive group. In the weight-multiplicity-free case, the classical medium, big and Kirillov algebras coincide. For the 56, we give three normalized gradient generators, their defining relations, and a polynomial formula for \(K_{27}\). A separate adjugate argument treats the nonminuscule \(14'\) of \(C_3\).