Classification of split reductive groups of semisimple rank one

By [F1] and [F3], $G^{\mathrm{\prime }}$ is split semisimple of rank one with split maximal torus $T_1=(T\cap G^{\mathrm{\prime }}{)}_t$. The exact adjoint quotient of [F2] is $q:G\to {\mathrm{P}\mathrm{G}\mathrm{L}}_2$ with kernel $Z(G)$. Since $G=Z(G{)}_t{G}^{\mathrm{\prime }}$ and $q$ kills the central torus, $q(G^{\mathrm{\prime }})={\mathrm{P}\mathrm{G}\mathrm{L}}_2$. The centre of semisimple $G^{\mathrm{\prime }}$ is finite by [F1], so the restriction $G^{\mathrm{\prime }}\to {\mathrm{P}\mathrm{G}\mathrm{L}}_2$ has finite central kernel and is a central isogeny. Choose $q$ up to an inner automorphism of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ so that $q(T)$ is its diagonal torus; split maximal tori of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ are conjugate over $k$, as proved in Milne20.31. The universal-cover input [F2] then lifts the standard ${\mathrm{S}\mathrm{L}}_2\to {\mathrm{P}\mathrm{G}\mathrm{L}}_2$ to a central isogeny $v_0:{\mathrm{S}\mathrm{L}}_2\to G^{\mathrm{\prime }}$ carrying $T_2$ onto $T_1$, as in the exact pair lift of Milne20.32. Composing with $G^{\mathrm{\prime }}\hookrightarrow G$ gives the required $v$. Its kernel is a subgroup of ${\mu }_2$, hence is $1$ or ${\mu }_2$; thus $G^{\mathrm{\prime }}$ is ${\mathrm{S}\mathrm{L}}_2$ or ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$. This is the central-cover proof, without invoking the later classification by arbitrary root data. It includes characteristic two and its nonreduced kernel.

For any homomorphism $v$ with central kernel, that kernel is a subgroup scheme of $Z({\mathrm{S}\mathrm{L}}_2)={\mu }_2$, hence is finite. Perfectness puts its image in $G^{\mathrm{\prime }}$, and equality of dimensions makes the image all of $G^{\mathrm{\prime }}$. Thus $v$ is a central isogeny, with kernel either $1$ or ${\mu }_2$. If $v,w$ are two such maps, their kernels agree and their induced identifications of the same central quotient with $G^{\mathrm{\prime }}$ differ by an automorphism; the universal central cover lifts this automorphism to ${\mathrm{S}\mathrm{L}}_2$. Since both maps carry $T_2$ onto $T_1$, the lift preserves $T_2$. Automorphisms of the split pair $({\mathrm{S}\mathrm{L}}_2,T_2)$ are precisely conjugations by elements of $(N_{{\mathrm{S}\mathrm{L}}_2}(T_2)\mathrm{/}{\mu }_2)(k)$, proving the asserted uniqueness.

The central quotient restricts to isomorphisms on the upper and lower unipotent subgroups: its kernel ${\mu }_2$ meets either subgroup scheme trivially, as is seen from the matrix coordinates, and the standard maps identify these subgroups with the two root groups of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$. Thus $v(U^{\pm })$ are the root groups of $(G,T)$, with the two signs possibly interchanged. The rank-one lattice $X_{\ast }(T_1)$ maps injectively to $\mathbb{Z}$ by $\lambda \mapsto ⟨\alpha ,\lambda ⟩$. For the chosen root $\alpha$, choose the sign of the standard cocharacter so that its image under $v$ has pairing $2$; it is the required ${\alpha }^{\vee }$, and injectivity proves uniqueness. In the simply connected case $\alpha =2\chi$ and ${\alpha }^{\vee }={\chi }^{\vee }$; in the adjoint case $\alpha =\chi$ and ${\alpha }^{\vee }=2{\chi }^{\vee }$. Since $U^{\pm }$ generate ${\mathrm{S}\mathrm{L}}_2$, their images generate $G^{\mathrm{\prime }}$. Finally $T=T_{1}Z(G{)}_t$ and $G=Z(G{)}_t{G}^{\mathrm{\prime }}$ by [F1] and [F3], so $T$ and the two root groups generate $G$.