Let \(E_0/F\) be the actual outer quadratic approximation field and \(B=\operatorname{End}_{E_0}(F^4)\). With lower odd critical level \(r=3h_1=2m+1\) and half leading level \(a=3h_2>r\), define
The filtered-centralizer estimate proves this is a compact group with \(J\) normal of index two. It does not assume \(E_0\subset K\).
For a new coset representative \(u\), the ratio \(\lambda=\theta^u/\theta\) on the actual \(H^1\) is nontrivial and sign-valued. Setting \(H_0=\ker\lambda\) gives
Its represented commutators are scalar; the quotient has exponent two; irreducibility and the exact scalar preimage prove nondegeneracy. The finite scalar center can exceed \(\mu_2\).
Scope
No literal extraspecial quotient is claimed. Full character values and marked attachment are retained through the actual finite extension.