The quotient of GL2 by the diagonal torus is the complement of the diagonal in P1 x P1

The closed subscheme of $G$ cut out by the two off-diagonal coordinates has $R$-points the invertible diagonal matrices, which form a subgroup of ${\mathrm{GL}}_2(R)$ for every $R$; by [F2] it is a closed subgroup scheme, and we call it $T$. As a scheme $T$ is the open subscheme $ad\mathrm{\ne }0$ of the affine plane with coordinates $a,d$, and by [F1] both $G$ and $T$ are smooth and affine of finite type over $k$.

By [F1] the standard representation identifies ${\mathrm{GL}}_V$ with $G$, and by [F3] the two factors $\mathbb{P}(V^{\vee })\cong {\mathbb{P}}^1$ carry the actions induced by it, whose product is the stated action of $G$ on $Y$; the point $o=(⟨e_1⟩,⟨e_2⟩)$ is a $k$-point of $Y$. A matrix $g=(ab;cd)$ preserves the line $⟨e_1⟩$ exactly when $c=0$ and preserves $⟨e_2⟩$ exactly when $b=0$, and the two stabilizer functors are closed; hence the scheme-theoretic stabilizer $G_o$ has $G_o(R)=T(R)$ for every $k$-algebra $R$ and equals $T$ by [F2].

Let $\mathrm{\Delta }\subseteq Y$ be the diagonal and define $q:G\to Y\setminus \mathrm{\Delta }$ by $q(g)=(g⟨e_1⟩,g⟨e_2⟩)$; it is well defined because $g$ is invertible and $⟨e_1⟩\mathrm{\ne }⟨e_2⟩$. For a field $K$, every $K$-point $(L_1,L_2)$ of $Y\setminus \mathrm{\Delta }$ has linearly independent generators $u\in L_1$, $v\in L_2$, and the matrix with columns $u,v$ is an invertible element of $G(K)$ mapping $o$ to $(L_1,L_2)$; hence $q$ is surjective on $K$-points and its image set is $Y\setminus \mathrm{\Delta }$.

Put $M=Y\setminus \mathrm{\Delta }$ and apply [F3] to its identity point, obtaining the two universal line subbundles $L_1,L_2\subseteq V_M$. On an open $U\subseteq M$ where both have frames $u,v$, the determinant $u\wedge v$ is nonzero in every residue field: two lines in a two-dimensional vector space are dependent exactly when they coincide, and the diagonal has been removed. Thus the determinant lies in no maximal ideal of any affine chart of $U$ and is a unit; the column matrix $a=(u,v)$ is invertible. It defines a section of $q$ over $U$ and proves $L_1\oplus L_2=V_U$. The isomorphism $U{\times }_{k}T\to q^{-1}(U)$, $(m,h)\mapsto a(m)h$, has inverse $g\mapsto (q(g),a(q(g){)}^{-1}g)$: the second component preserves the two coordinate lines and hence belongs to the diagonal torus by step 1.2. These opens cover $M$, so $q$ is locally a projection with fibre the torus $T$, flat and locally of finite presentation by [F1] and [F5]. It is surjective by step 1.3, hence faithfully flat.

The morphism of the criterion $G{\times }_{k}T\to G{\times }_{Y\setminus \mathrm{\Delta }}G$, $(g,h)\mapsto (g,gh)$, is an isomorphism: on $R$-points for every $k$-algebra $R$ it is a bijection onto the pairs $(g,g^{\mathrm{\prime }})\in G(R{)}^2$ with $q(g)=q(g^{\mathrm{\prime }})$, with inverse $(g,g^{\mathrm{\prime }})\mapsto (g,g^{-1}{g}^{\mathrm{\prime }})$, and $g^{-1}{g}^{\mathrm{\prime }}\in T(R)$ exactly when $q(g)=q(g^{\mathrm{\prime }})$ by the stabilizer computation of step 1.2.

Applying the criterion [F4] to $U=G$, $R=G{\times }_{k}T$ with $s(g,h)=g$, $t(g,h)=gh$, $M=Y\setminus \mathrm{\Delta }$ and $q$ from steps 2.1 and 2.2 shows that $Y\setminus \mathrm{\Delta }$ represents the fppf quotient sheaf $G\mathrm{/}T$ and that the quotient morphism is $q$. Since the image of $q$ is $Y\setminus \mathrm{\Delta }$ by step 1.3, the orbit subscheme $O_o$ and $Y\setminus \mathrm{\Delta }$ agree; so $O_o$ represents $G\mathrm{/}T$, the quotient morphism is ${\varrho }_o=q$, and the conclusions of (a) and (b) follow, including the representability of $G\mathrm{/}T$.

For (c): the quotient $G\mathrm{/}T$ is isomorphic to $O_o$ by step 3.1; the orbit $O_o$ is smooth over $k$ and of finite type by Smooth orbits are locally closed and their orbit maps are faithfully flat over every field, and it is separated over $k$ because it is a locally closed subscheme of the separated finite-type $k$-scheme $Y$ by [F5], an immersion being separated (Open and closed immersions are separated) and separatedness being stable under composition (Separated morphisms compose). Base changing to an algebraic closure $\bar{k}$, the orbit-stabilizer identity [F6] applied to the connected smooth group $G_{\bar{k}}$ acting on $Y_{\bar{k}}$ and the orbit $O_{o,\bar{k}}$ gives $\dim O_{o,\bar{k}}=\dim G_{\bar{k}}-\dim T_{\bar{k}}=4-2=2$, and $(G\mathrm{/}T{)}_{\bar{k}}\cong O_{o,\bar{k}}$, which is the stated dimension. Here ${\mathrm{GL}}_2$ is the nonempty open subset $d\mathrm{\ne }0$ of ${\mathbf{A}}^4$, of dimension $4$ by [F5] and Affine and projective n-space have dimension n, and $T$ is the nonempty open subset $ad\mathrm{\ne }0$ of ${\mathbf{A}}^2$, of dimension $2$ by the same two results; $G_{\bar{k}}$ is connected because it is a nonempty open subscheme of the irreducible ${\mathbf{A}}_{\bar{k}}^4$, whose coordinate ring $\bar{k}[x_1,\dots ,x_4]$ is a domain (A polynomial ring over an integral domain is an integral domain), so that the zero ideal corresponds to ${\mathbf{A}}^4$ under the Nullstellensatz correspondence (Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals) and nonempty open subschemes are irreducible by Irreducibility via nonempty open subsets, connectedness and open subspaces.

The first projection $Y\setminus \mathrm{\Delta }\to {\mathbb{P}}^1$ is the composite of the isomorphism $G\mathrm{/}T\cong Y\setminus \mathrm{\Delta }$ with ${\mathrm{p}\mathrm{r}}_1$; over a point $y$ put $K=\kappa (y)$ and let $L_1$ be its canonical $K$-point; the fibre is $\{L_2\in {\mathbb{P}}_K^1:L_2\mathrm{\ne }{L}_1\}$, which is ${\mathbb{P}}_K^1$ with one closed point removed. Finally the natural map $G(K)\mathrm{/}T(K)\to (G\mathrm{/}T)(K)$ is surjective because every $K$-point of $Y\setminus \mathrm{\Delta }$ is $q(g)$ for some $g\in G(K)$ by step 1.3, and injective because $q(g)=q(g^{\mathrm{\prime }})$ forces $g^{-1}{g}^{\mathrm{\prime }}\in T(K)$ by step 2.2; hence it is a bijection for every extension field $K\mathrm{/}k$. This completes the verification of (c).