Three Quadratics and a Cubic: Triality, the Racah Algebra and the Dual Pair D₄×D₄ in E₈

We determine the invariant algebra of the dual pair of type D₄ × D₄ on the minimal nilpotent orbit of E₈ and its quantization in the Joseph quotient. Triality supplies three quadratic invariants and a cubic whose Poisson brackets form the Racah algebra. An explicit transverse root subsystem proves generation; the total invariant algebra is polynomial in degrees 2, 2, 2, 3, 4 and 4, and its central fibres form a miniversal deformation of the D₄ Kleinian surface. We identify every quantum central quotient with a deformed preprojective corner, classify its four finite-dimensional simple modules at dominant integral characters, and determine the unitary Wilson transform on each compact branching ladder. A nullcone calculation proves covariant freeness and infinite compact multiplicities at every central character. We obtain infinite-length spherical theta lifts for the split real pair and infinite multiplicity for regular tempered spherical representations over characteristic-zero nonarchimedean local fields. For A₄ × A₄, we construct a flat graded Poisson deformation of the E₈ Kleinian surface and prove agreement of its total algebra with the invariant ring through degree 31. We determine its quadratic deformation parameter exactly; global generation and the proposed full Weyl symmetry remain conjectural.