I observe that in the "standard finite point group" symmetries containing higher dimensional irreducible representations (over $\mathbb{R}$), the total product of the highest dimensional representation(s) with itself always contains the irreducible representation wich transforms like the rotation around the (main) axis of symmetry.
My question is, if there is any argument by which that can be shown other than simply calculating it for each point group separately. Or so to say "Why is it like that?".
To make the observation clear I give some examples (using Schönflies notation):
Take point group O$_h$, the highest dimensional irreps are T$_{1g}$, T$_{2g}$, T$_{1u}$, T$_{2u}$, the rotation around (in this case any) axis transforms in this group like T$_{1g}$, and we have:
$$\mathrm{T}_{1g}\otimes\mathrm{T}_{1g} = \mathrm{A}_{1g} ⊕ \mathrm{E}_g ⊕ \mathrm{T}_{1g} ⊕ \mathrm{T}_{2g}$$
which contains $\mathrm{T}_{1g}$.
the same holds for the other three, in fact in this case: $$\mathrm{T}_{2g}\otimes\mathrm{T}_{2g} = \mathrm{T}_{1u}\otimes\mathrm{T}_{1u} = \mathrm{T}_{2u}\otimes\mathrm{T}_{2u} = \;\;\;\;\mathrm{A}_{1g} ⊕ \mathrm{E}_g ⊕ \mathrm{T}_{1g} ⊕ \mathrm{T}_{2g}$$
Lets take D$_{4h}$. Here the highest dimensional irrep are E$_g$ and E$_u$, and the Rotation around the main axis is A$_{2g}$. And we have:
$$\mathrm{E}_{g}\otimes\mathrm{E}_{g} = \mathrm{A}_{1g} ⊕ \mathrm{A}_{2g} ⊕ \mathrm{B}_{1g} ⊕ \mathrm{B}_{2g},$$ which again contains A$_{2g}$.
Another though quite related observation is that, when one distorts the symmetry along one coordinate which is represented by an irrep contained it the decomposition (but not the total symmetric one and not the one representing the rotation around the main axis) one yields a subgroup and a splitting of the degenerate irreps into new irreps of the new subgroup. But again the total product of the new, now different irreps contains the irrep representing the rotation around the main axis.
Example: D$_{4h}$ is distorted along B$_{1u}$ to yield D$_{2h}$, one of the two highest degenerate irreps in D$_{4h}$ is E$_g$, upon distortion to D$_{2h}$, E$_g$ splits into B$_{2g}$ and B$_{3g}$. B$_{2g} \otimes$ B$_{3g}$ = B$_{1g}$. Again the rotation around the main axis in D$_{2h}$ transforms like B$_{1g}$.
Except checking everything case by case, I have no idea how to show why this happens. Any ideas appreaciated!