| Summary: | The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>3</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula>. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of <i>k</i> orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>3</mn></msub></semantics></math></inline-formula>. The geometrical structures of nested polytopes are exemplified.
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