Cartan subgroup (reviewed)

Let $R$ be a commutative ring. Given a free rank $2$ étale $R$-algebra $A$ equipped with a basis, any $a \in A^\times$ defines an $R$-linear multiplication-by-$a$ map $A \to A$, so we get an injective homomorphism $A^\times \to \Aut_{\text{$R$-module}}(A) \simeq \GL_2(R)$, and the image is called a Cartan subgroup of $\GL_2(R)$. The canonical involution of the $R$-algebra $A$ gives another element of $\Aut_{\text{$R$-module}}(A)$; we call the group generated by it and the Cartan subgroup $A^\times$ the extended Cartan subgroup. The Cartan subgroup has index $2$ in the extended Cartan subgroup.

If $R=\F_p$, there are two possibilities for $A$: the split algebra $\F_p \times \F_p$ and the nonsplit algebra $\F_{p^2}$; the resulting Cartan subgroups are called split and nonsplit. The extended Cartan subgroup equals the normalizer of the Cartan subgroup in $\GL_2(\F_p)$ except when $p=2$ and $A$ is split. In the split case, if we use the standard basis of $\F_p \times \F_p$, the Cartan subgroup is the subgroup of diagonal matrices in $\GL_2(\F_p)$, and the extended Cartan subgroup is this together with the coset of antidiagonal matrices in $\GL_2(\F_p)$.

If $R=\Z/p^e\Z$, again there are two possibilities for $A$: the split algebra $R \times R$, or the nonsplit algebra. The nonsplit algebra can be described as $\mathcal{O}/p^e \mathcal{O}$ where $\mathcal{O}$ is either the degree $2$ unramified extension of $\Z_p$ or a quadratic order in which $p$ is inert. The nonsplit algebra can also be described as the ring of length $e$ Witt vectors $W_e(\F_{p^2})$.

If $R=\Z/N\Z$ for some $N \ge 1$, then $A$ can be split or nonsplit independently at each prime dividing $N$.