Remark. what the real quaternion example settles [fgap-001W]

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The general route has several distinct endpoints. Over an algebraically closed field, Wedderburn blocks and the character table organize all simple representations. Over \(\mathbb R\), division-algebra type and descent data enter. For a normal subgroup, Clifford theory organizes restriction and induction. With a nontrivial factor set, projective representations are modules over a twisted algebra.

The quaternion calculation in the real Group Algebra of the quaternion group closes 1 bounded endpoint: it gives an explicit real-algebra equivalence \(\mathbb R[Q_8]\cong \mathbb R^4\times \mathbb H\), and Real blocks of the quaternion Group Algebra identifies its center. It does not prove a general real Wedderburn theorem, the characters of \(2T\), or the Clifford-theory extension step. Those are separate statements and do not follow from the worked example.