Let \(\Gamma =Q_8=\{\pm 1,\pm i,\pm j,\pm k\}\) and take the simple undirected Cayley graph with edges \(\{g,gs\}\), \(s\in \{\pm i,\pm j\}\), counted once each. Give each edge weight one, choose any orientation, and set \(m=1\). The two parts \[A=\{\pm 1,\pm k\},\qquad B=\{\pm i,\pm j\}\] give \(K_{4,4}\): every vertex in one part has the 4 distinct vertices of the other as its neighbors. In part order, \[L=\begin {pmatrix}4I_4&-J_4\\-J_4&4I_4\end {pmatrix}, \qquad Q=L+I_8,\] where \(J_4\) is the all-ones matrix. The two constant-on-part vectors give Laplacian eigenvalues \(0,8\); the two partwise zero-sum spaces give eigenvalue \(4\), of total multiplicity 6.

Left multiplication gives a signed graph action \(\rho \) on \(\mathbb R^{Q_8}\), independent of the arbitrary edge orientations. For the central involution \(z=-1\), put \[e_-=(1-[z])/2\in \mathbb R[Q_8],\qquad P_-=\rho (e_-)=(I-\rho (z))/2.\] Since \(z^2=1\) and \(z^{-1}=z\), this is a self-adjoint idempotent. Its image is the four-dimensional space of fields \(f(-g)=-f(g)\). The group-algebra decomposition \(\mathbb R[Q_8]\cong \mathbb R^4\times \mathbb H\) in Proposition [fgap-001B] sends \(e_-\) to \((0,1)\): each real character takes \(z\) to \(1\), while the quaternionic realization takes it to \(-1\). Thus this selector retains precisely the quaternion factor in the regular module, without assigning it a physical interpretation.

Compare the coordinate selector on the subgroup \(R=\langle i\rangle =\{1,-1,i,-i\}\), which retains two vertices from each part. In the full graph, decompose fields into the two constant-on-part lines and the two partwise zero-sum spaces. On the normalized part constants, the precision and covariance are \[\begin {pmatrix}5&-4\\-4&5\end {pmatrix},\qquad \frac 19\begin {pmatrix}5&4\\4&5\end {pmatrix};\] on the zero-sum spaces the covariance is \(1/5\). It follows that full covariance entries are \(13/45\) on the diagonal, \(4/45\) between distinct vertices in the same part, and \(1/9\) between parts. For example the first entry is \((1-1/4)/5+(5/9)/4=13/45\).

In retained order \((1,-1,i,-i)\), compression therefore gives \[C_R=\frac 1{45}\begin {pmatrix} 13&4&5&5\\4&13&5&5\\5&5&13&4\\5&5&4&13 \end {pmatrix}.\] The two within-part difference vectors have covariance eigenvalue \((13-4)/45=1/5\). On normalized retained part constants the covariance block is \(\frac 1{45}\begin {pmatrix}17&10\\10&17\end {pmatrix}\), with eigenvalues \(3/5,7/45\). Inversion gives the retained precision spectrum \[\operatorname {spec}(S_R)=\{5/3,5,5,45/7\}.\] It differs from \(\{5,5,5,5\}\) for the quaternion selector despite equal rank and the same full background precision.

The chosen \(Q_8\) action is auxiliary data used to construct \(P_-\); it is not another slot of the selected tuple. Indeed the underlying graph has \(2(4!)^2=1152\) automorphisms: its connected bipartition is unique up to exchange, and every permutation within either part, as well as their exchange, preserves all edges. The selected-background groupoid remembers the graph and the resulting projector, not a named subgroup of this automorphism group. Remembering the action would require an enhanced tuple and arrows intertwining that action.