Rank-one connected groups

Work over an algebraic closure. Set $S=G\mathrm{/}R(G)$. By the radical-quotient theorem cited in [F1], $S$ is semisimple and its Borel variety identifies with that of $G$. The rank-one theorem cited in [F1] then gives the equivalence: in rank $1$ the Weyl group of $S$ acts faithfully and transitively on the Borel subgroups containing $\bar{T}$, while the Borel-opposition lemma cited in [F1] distinguishes the positive and negative Borels; conversely, the two $T$-fixed points of the Borel variety and the fixed-point bound in [F3] give dimension at most $1$, while nonsolvability excludes dimension zero; and a one-dimensional complete homogeneous curve is ${\mathbf{P}}^1$, whose action gives the isogeny to ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$. An isogeny to ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ forces rank $1$.

Since $S$ is isogenous to ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$, it has dimension $3$, semisimple rank $1$, and root system $\{\alpha ,-\alpha \}$. The root decomposition is taken in $\mathrm{Lie}(S)$, not in $\mathrm{Lie}(G)$: over the algebraic closure it is $\mathrm{Lie}(\bar{T})\oplus {\mathfrak{s}}_{\alpha }\oplus {\mathfrak{s}}_{-\alpha }$, with each root space one-dimensional.

The quotient map identifies $G\mathrm{/}B$ with $S\mathrm{/}\bar{B}$, so this is ${\mathbf{P}}^1$ by step 1.1. The action of $G$ on $G\mathrm{/}B$ factors through $S$; it is transitive on ${\mathbf{P}}^1$, whereas a connected solvable affine group has a fixed point on a complete variety. Thus its image is nonsolvable, and every proper connected subgroup of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ is solvable, so the action map is surjective. Its kernel in $S$ is $Z(S)$ by [F3], so its kernel in $G$ is exactly $q^{-1}(Z(S))$.

The universal central cover in [F2] lifts the central isogeny of step 2.2 to a central isogeny ${\mathrm{S}\mathrm{L}}_2\to S$, carrying the diagonal Borel, upper root group and nontrivial Weyl representative to $\bar{B},\bar{U},\bar{n}$. The elementary matrix decomposition ${\mathrm{S}\mathrm{L}}_2=B_2\bigsqcup U_2^{+}{n}_2{B}_2$ separates matrices by whether their lower-left entry is zero. Its images give $S=\bar{B}\bigsqcup \bar{U}\bar{n}\bar{B}$; disjointness follows from the two orbits on $S\mathrm{/}\bar{B}={\mathbf{P}}^1$.

Finally let $G$ be split reductive of total rank $1$ and semisimple rank $1$. Its largest central torus has dimension zero, hence is trivial; by the centre structure supplier $G$ is semisimple. Milne's split rank-one adjoint result in [F2] gives a central isogeny $q:G\to {\mathrm{P}\mathrm{G}\mathrm{L}}_2$ with kernel $Z(G)$. Universality of the standard ${\mathrm{S}\mathrm{L}}_2$ cover in [F2] supplies a central isogeny $f:{\mathrm{S}\mathrm{L}}_2\to G$ over $k$ lifting it. Its kernel is a subgroup scheme of $\ker (qf)={\mu }_2$, hence is either $1$ or ${\mu }_2$ (including characteristic $2$). Consequently $G$ is ${\mathrm{S}\mathrm{L}}_2$ or ${\mathrm{S}\mathrm{L}}_2\mathrm{/}{\mu }_2={\mathrm{P}\mathrm{G}\mathrm{L}}_2$, respectively. This proves the final classification without a later general root-datum classification theorem.