$ as a $\kappa$-variety so that the number of rational points is $2f$, where $f$ is the cardinality of $\kappa$.

Construction following our technique {#cfot}
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Let $q$ be the function defined over $L$ such that $$q : L\longrightarrow A, l\mapsto h(l,l).$$ If we write $l=x+\pi y$ such that $x, y \in A$, then $q(l)=h(x+\pi y, x+\pi y)=x^2-2\delta y^2$. Thus $q$ mod 2 is an additive polynomial over $\kappa$. Let $B(L)$ be the sublattice of $L$ such that $B(L)/\pi L$ is the kernel of the additive polynomial $q$ mod 2 on $L/\pi L$. In this case, $B(L)=\pi L$.

We define another sublattice $Z(L)$ of $L$ such that $Z(L)/\pi B(L)$ is the radical of the quadratic form $\frac{1}{2}q$ mod 2 on $B(L)/\pi B(L)$. In this case, $Z(L)=\pi B(L)=2L$.

For an étale $A$-algebra $R$ with $g\in \mathrm{Aut}_{B\otimes_AR}(L\otimes_AR, h\otimes_AR)$, it is easy to see that $g$ induces the identity on $L/B(L)=L/\pi L$. An element $g$ also induces the identity on $L/Z(L)=L/2L$ since $$q(gw-w, gw-w)=2q(w)-\left(h(gw, w)+h(w, gw)\right)=$$ $$2q(w)-\left(h(w+x, w)+h(w, w+x)\right) =-\left(h(x, w)+h(w, x)\right)$$ where $w\in L$ and $x\in B(L)=\pi L$ such that $gw=w+x$, and thus $h(x, w)\in \pi B$ and $\left(h(x, w)+h(w, x)\right)\in 4 B$.

Based on this, we construct the following functor from the category of commutative flat $A$-algebras to the category of monoids as follows. For any commutative flat $A$-algebra $R$, set $$\underline{M}(R) = \{m \in \mathrm{End}_{B\otimes_AR}(L \otimes_A R)\} ~|~ \textit{$m$ induces the identity on $L\otimes_A R/ Z(L)\otimes_A R$}\}.$$ This functor $\underline{M}$ is then representable by a polynomial ring and has the structure of a scheme of monoids. Let $\underline{M}^{\ast}(R)$ be the set of invertible elements in $\underline{M}(R)$ for any commutative $A$-algebra $R$. Then $\underline{M}^{\ast}$ is representable by a group scheme which is an open subscheme of $\underline{M}$ (Section \[m\]). Thus $\underline{M}^{\ast}$ is smooth. Indeed, in our case, $\underline{M}^