^2-2b^2=1\}$ for a flat $A$-algebra $R$. Thus we cannot guarantee that $a-1$ is contained in the ideal $(2)$, which should be necessary in order that $(a, b)$ is an element of $\underline{G}(R)$.

The special fiber of the smooth integral model {#sf}
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In this section, we will determine the structure of the special fiber $\tilde{G}$ of $\underline{G}$ by determining the maximal reductive quotient and the component group when $E/F$ satisfies *Case 2*, by adapting the approach of Section 4 of [@C1] and Section 4 of [@C2]. From this section to the end, the identity matrix is denoted by id.

The reductive quotient of the special fiber {#red}
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Assume that $i=2m$ is even. Recall that $Z_i$ is the sublattice of $B_i$ such that $Z_i/\pi B_i$ is the radical of the quadratic form $\frac{1}{2^{m+1}}q$ mod 2 on $B_i/\pi B_i$, where $\frac{1}{2^{m+1}}q(x)=\frac{1}{2^{m+1}}h(x,x)$.

\[t43\] Assume that $i=2m$ is even. Let $\bar{q}_i$ denote the nonsingular quadratic form $\frac{1}{2^{m+1}}q$ mod 2 on $B_i/Z_i$. Then there exists a unique morphism of algebraic groups $$\varphi_i:\tilde{G}\longrightarrow \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$$ defined over $\kappa$, where $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$ is the reduced subgroup scheme of $\mathrm{O}(B_i/Z_i, \bar{q}_i)$, such that for all étale local $A$-algebras $R$ with residue field $\kappa_R$ and every $\tilde{m} \in \underline{G}(R)$ with reduction $m\in \tilde{G}(\kappa_R)$, $\varphi_i(m)\in \mathrm{GL}(B_i\otimes_AR/Z_i\otimes_AR)$ is induced by the action of $\tilde{m}$ on $L\otimes_AR$ (which preserves $B_i\otimes_AR$ and $Z_i\otimes_AR$ by the construction of $\underline{M}$).

The proof of this theorem is similar to that of Theorem 4.3 of [@C2]. Thus we only provide the image of an element $m$ of $\tilde{G}(\kappa_R)$ in $\mathrm{O}(A_i/Z_i, \bar{q}_i)_{\mathrm{red}}(\kappa_R)$, where $R$ is an étale local $A$-algebra with $\kappa_R$ as its residue field.
