proposition. the real Group Algebra of the quaternion group [fgap-001B]
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    Let \(Q_8=\{\pm 1,\pm i,\pm j,\pm k\}\) be the concrete quaternion group from the quaternion group inside Hamilton's Quaternions, and put \[ \mathcal {E}=\{-1,1\}\subset \mathbb {R}^{\times }, \qquad \mathcal {X}=\mathcal {E}\times \mathcal {E}. \] For \((\epsilon ,\delta )\in \mathcal {X}\), there is a real character \[ \chi _{\epsilon ,\delta }:Q_8\longrightarrow \mathbb {R}^{\times } \] determined by \[ \chi _{\epsilon ,\delta }(-1)=1,\qquad \chi _{\epsilon ,\delta }(i)=\epsilon ,\qquad \chi _{\epsilon ,\delta }(j)=\delta ,\qquad \chi _{\epsilon ,\delta }(k)=\epsilon \delta . \] These are the four characters pulled back from \(Q_8/\{\pm 1\}\cong C_2\times C_2\); compare [etingof2024introduction, sec. 4.3, pp. 63--64 and ex. 4.8.1, p. 73].

    Write \(\mathbb {R}^{\mathcal {X}}\) for the algebra of functions \(\mathcal {X}\to \mathbb {R}\), with pointwise operations. Extending the four characters and the inclusion \(Q_8\subset \mathbb {H}^{\times }\) linearly gives an algebra homomorphism \[ \Phi :\mathbb {R}[Q_8]\longrightarrow \mathbb {R}^{\mathcal {X}}\times \mathbb {H}. \] Then \(\Phi \) is an isomorphism of real algebras. The displayed map retains the chosen character index and quaternionic realization as part of its data.

    Proof. 

      Let \(u_q\) be the Group-Algebra basis element indexed by \(q\in Q_8\).

      Write a general element in the form \[ x=\sum _{q\in Q_8}a_qu_q. \] For \(q\in \{1,i,j,k\}\), set \[ s_q=a_q+a_{-q},\qquad d_q=a_q-a_{-q}. \] The quaternion coordinate of \(\Phi (x)\) is \[ d_1+d_i i+d_j j+d_k k. \] It therefore recovers the four differences \(d_q\).

      The coordinate indexed by \((\epsilon ,\delta )\in \mathcal {X}\) is \[ y_{\epsilon ,\delta } =s_1+\epsilon s_i+\delta s_j+\epsilon \delta s_k. \] The inverse Hadamard calculation gives \[ \begin {aligned} s_1&=\frac 14\sum _{(\epsilon ,\delta )\in \mathcal {X}}y_{\epsilon ,\delta },& s_i&=\frac 14\sum _{(\epsilon ,\delta )\in \mathcal {X}} \epsilon y_{\epsilon ,\delta },\\ s_j&=\frac 14\sum _{(\epsilon ,\delta )\in \mathcal {X}} \delta y_{\epsilon ,\delta },& s_k&=\frac 14\sum _{(\epsilon ,\delta )\in \mathcal {X}}\epsilon \delta y_{\epsilon ,\delta }. \end {aligned} \] Thus the four real coordinates recover the four sums \(s_q\). Finally, \[ a_q=\frac {s_q+d_q}{2},\qquad a_{-q}=\frac {s_q-d_q}{2}, \] so \(\Phi (x)\) recovers all eight coefficients of \(x\). Hence \(\Phi \) is injective. Its domain and codomain both have real dimension 8, so it is bijective.

      The quaternion coordinate alone is surjective, because its basis values include \(1,i,j,k\). It vanishes exactly when \(d_1=d_i=d_j=d_k=0\), or equivalently when \(a_q=a_{-q}\) for \(q\in \{1,i,j,k\}\). Its kernel is therefore the 4-dimensional span of \[ u_1+u_{-1},\quad u_i+u_{-i},\quad u_j+u_{-j},\quad u_k+u_{-k}. \] It therefore induces \[ \mathbb {R}[Q_8]/\ker (\Phi _{\mathbb {H}}) \cong _{\mathbb {R}\text {-alg}}\mathbb {H}. \] This quotient depends on the chosen quaternionic realization. The combined map \(\Phi \) is what exhibits the selected quotient as the quaternion factor of the displayed product; a quotient map by itself does not supply that direct-factor statement.

      The quaternion model is grounded in [voight2021quaternion, sec. 11.2, p. 166]. The extension of group representations to Group-Algebra maps follows [sengupta2010representations, secs. 3.1--3.2, pp. 39--41]. The coefficient recovery above is direct and does not assume a general classification theorem. No physical meaning is assigned to a factor merely from its dimension or familiar algebra.