Algebraically Independent Generators for the Invariant Field of \({\textrm{SO}_{3}({\mathbb{R}})}\) and \({\textrm{O}_{3}({\mathbb{R}})}\) Representations \({{\mathbb{R}}^3\oplus {\mathcal{H}}}\)

Abstract. We consider the representations of the group [Formula: see text] that contain the standard representation [Formula: see text] as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors, or the classical product space [Formula: see text] which appears in multiple physical and engineering situations. For any such representation, we construct a set of algebraically independent rational invariants that generates the invariant field. Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the [Formula: see text]-invariant field on the full representation [Formula: see text] with the invariant field of a representation of [Formula: see text] on a subspace [Formula: see text]. We exhibit a set of [Formula: see text]-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational [Formula: see text]-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of [Formula: see text] and several cases of interest to applications are detailed as examples.