  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 $H_2$, $H_3$ and $H_4$.
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 $k$ 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 $H_2$ and $H_3$. The geometrical structures of nested
polytopes are exemplified.
