${\widetilde{\mathcal{I}}}_{{\mathcal{G}}}$, i.e. $\widetilde{{\mathcal{G}}} = \widetilde{{\mathcal{I}}} \cap \widetilde{{\mathcal{I}}}_{{\mathcal{G}}}$. The matrix $R$ in , which reduces ${\widetilde{\mathcal{I}}}$ into irreps as in , decomposes the representation ${\widetilde{\mathcal{G}}}$ as follows: $$\label{g_rep}
\hat{{\mathcal{G}}}:= R^{-1}{\widetilde{\mathcal{G}}}R = {\mathcal{G}}_1 \oplus {\mathcal{G}}_2,$$ where ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ are matrix subgroups of the irreps $\rho_3$ and $\rho_3'$ given in Table \[irreps\_ico\], respectively. Notice that ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ are not necessarily irreducible representations of ${\widetilde{\mathcal{G}}}$.

The matrix $R$ in general does not reduce the representation ${\widetilde{\mathcal{I}}}_{{\mathcal{G}}}$, since the subspaces $E^{\parallel}$ and $E^{\perp}$, which are invariant under ${\widetilde{\mathcal{I}}}$, are not necessarily invariant under ${\widetilde{\mathcal{I}}}_{{\mathcal{G}}}$. Let us denote by $R_{{\mathcal{G}}} \in O(6)$ the orthogonal matrix that reduces ${\widetilde{\mathcal{I}}}_{{\mathcal{G}}}$ into irreps, i.e. $$\hat{{\mathcal{I}}}_{{\mathcal{G}}} := R_{{\mathcal{G}}}^{-1} {\widetilde{\mathcal{I}}}_{{\mathcal{G}}} R_{{\mathcal{G}}} \simeq T_1 \oplus T_2,$$ where $T_1$ and $T_2$ are the two non-equivalent three-dimensional irreps of ${\mathcal{I}}$. This matrix carries the bases of a physical and a parallel space which are invariant under ${\widetilde{\mathcal{I}}}_{{\mathcal{G}}}$. We denote these spaces by $E^{\parallel}_{{\mathcal{G}}}$ and $E^{\perp}_{{\mathcal{G}}}$, respectively, and we write $\pi^{\parallel}_{{\mathcal{G}}}$ and $\pi^{\perp}_{{\mathcal{G}}}$ for the corresponding projections. By , we have $$R_{{\mathcal{G}}}^{-1} = \left( \begin{array}{c}
\pi^{\parallel}_{{\mathcal{G}}} \\
\pi^{\perp}_{{\mathcal{G}}} 
\end{array} \right).$$

The matrix $R_{{\mathcal{G}}}$ is in general not unique. With a suitable choice of the basis vectors constituting the columns of $R_{{\mathcal{G}}}$, we assume 