Remark. the center is a commutative shadow [fgap-000O]

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The quaternion group \(Q_8\) has a real Group Algebra decomposition \[ \mathbb {R}[Q_8]\cong \mathbb {R}^4\times \mathbb {H}. \] This can be seen directly from the concrete copy \(Q_8\subset \mathbb {H}^{\times }\) in the quaternion group inside Hamilton's Quaternions, rather than assumed from a classification theorem.

Let \(u_q\) denote the basis element of the Group Algebra indexed by \(q\in Q_8\), and write a general element as \(x=\sum _{q\in Q_8}a_qu_q\). The four characters of \(Q_8/\{\pm 1\}\cong C_2\times C_2\) and the quaternionic map \(u_q\mapsto q\) combine to an algebra homomorphism \[ \Phi :\mathbb {R}[Q_8]\longrightarrow \mathbb {R}^4\times \mathbb {H}. \] For \(q\in \{1,i,j,k\}\), put \(s_q=a_q+a_{-q}\) and \(d_q=a_q-a_{-q}\). The quaternion coordinate of \(\Phi \) recovers \[ d_1+d_i i+d_j j+d_k k. \] The four real coordinates recover the Hadamard transform \[ s_1+\epsilon s_i+\delta s_j+\epsilon \delta s_k, \qquad \epsilon ,\delta \in \{\pm 1\}. \] The Hadamard matrix is invertible, so these four values recover every \(s_q\). Together with the four differences, they recover every coefficient. Thus \(\Phi \) has zero kernel. Both sides have real dimension 8, so \(\Phi \) is an isomorphism. The quaternionic realization and multiplication used here are developed in Hamilton's Quaternions and [voight2021quaternion, sec. 11.2, p. 166].

Taking centers now gives \[ Z\bigl (\mathbb {R}[Q_8]\bigr ) \cong Z\bigl (\mathbb {R}^4\times \mathbb {H}\bigr ) \cong \mathbb {R}^5. \] The comparison is: \[ \begin {array}{c|c|c} \text {object}&\text {retained data}&\text {forgotten data}\\ \hline \mathbb {R}[Q_8]&4\mathbb {R}\text { and }\mathbb {H}&\text {none here}\\ Z(\mathbb {R}[Q_8])&\text {five central factors}& \mathbb {H}\text { and block size or type}. \end {array} \] For a real-centered simple block, the center alone cannot distinguish matrix size or real from quaternionic type. A complex block still has complex center. Hence \(\operatorname {Spec}Z(\mathbb {R}[Q_8])\) is a selected commutative shadow, not a lossless localization of the noncommutative algebra.