The opposite-root big cell is an open chart

The varieties $U^{-}\cong {\mathbb{A}}^m$, $T\cong {\mathbb{G}}_m^l$ and $U\cong {\mathbb{A}}^m$ are smooth over $\mathbb{C}$ by [F1], so $X:=U^{-}\times T\times U$ is smooth over $\mathbb{C}$ of dimension $2m+l=\dim G$; in particular the local rings of $X$ and of $G$ at closed points are regular local rings of the same dimension. At the origin $(1,1,1)$ the tangent space is the direct sum ${\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}=\mathfrak{g}$ by [F2], and the differential $\mathrm{d}{m}_{(1,1,1)}$ is the sum map $(X,Y,Z)\mapsto X+Y+Z$; it is therefore a linear isomorphism.

$m$ is injective on $R$-points for every $\mathbb{C}$-algebra $R$. First put $H=U^{-}\cap B$, a closed finite-type subgroup scheme. Its Lie algebra is ${\mathfrak{n}}^{-}\cap \mathfrak{b}=0$ by [F2]; hence its local ring at the identity has zero cotangent space and is a field by Nakayama. Translation gives the same at every closed point, so $H$ is zero-dimensional and reduced, hence finite étale over $\mathbb{C}$. Every element of $H(\mathbb{C})$ is a finite-order element of the unipotent group $U^{-}$ from [F1]; in a faithful matrix representation a finite-order unipotent matrix in characteristic zero is the identity. Thus $H(\mathbb{C})=\{1\}$, and reducedness gives $H=1$ as a group scheme. Now if $b\in (B\cap B^{-})(R)$ for any $\mathbb{C}$-algebra $R$, write $b=t{u}^{-}$ with $t\in T(R)$ and $u^{-}\in U^{-}(R)$ using $B^{-}=T⋉U^{-}$; since $b,t\in B(R)$, the element $u^{-}$ lies in $H(R)=1$. Therefore $b=t$ and $B\cap B^{-}=T$ as closed subgroup schemes. Now let $x=(u^{-},t,u)$ and $y=(u_1^{-},t_1,u_1)$ in $G(R)$ with $x=y$ in $G(R)$, i.e. $u^{-}tu=u_1^{-}{t}_1{u}_1$. Then $(u_1^{-}{)}^{-1}{u}^{-}\cdot t=t_1\cdot u_1{u}^{-1}$; the left side is an $R$-point of $B^{-}$ and the right side an $R$-point of $B$, so both are $R$-points of $B\cap B^{-}=T$. Thus $(u_1^{-}{)}^{-1}{u}^{-}=t{t}^{\mathrm{\prime }-1}\in T(R)$ for some $t^{\mathrm{\prime }}\in T(R)$ and also lies in $U^{-}(R)$, so $(u_1^{-}{)}^{-1}{u}^{-}=1$ because $T\cap U^{-}=1$ (the negative-root case of [F1]); and $u_1{u}^{-1}\in T(R)\cap U(R)=1$, so $u_1=u$, after which $t_1=t$ from the equation. Hence $m(R)$ is injective for every $R$.

The differential of $m$ is invertible at every closed $\mathbb{C}$-point of $X$. Indeed $m(u^{-}\delta ,t,yu)=u^{-}\cdot m(\delta ,t,y)\cdot u$, so left and right translations reduce the assertion to $(1,t,1)$. There $m(\delta ,t\tau ,y)=t\cdot {\mathrm{Ad}}_{t^{-1}}(\delta )\cdot \tau \cdot y$, whose differential, after left translation by $t^{-1}$ in $G$, is the direct-sum map ${\mathrm{Ad}}_{t^{-1}}{\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}\to \mathfrak{g}$. Since $T$ preserves each root space by [F1] and [F2], this map is an isomorphism by step 1.1.

$m$ is flat. First let $p$ be a closed $\mathbb{C}$-point of $X$ and $q=m(p)$, also a closed $\mathbb{C}$-point. The local rings ${\mathcal{O}}_{G,q}$ and ${\mathcal{O}}_{X,p}$ are regular of the same dimension $\dim G$ by step 1.1; the map on their cotangent spaces is the dual of the isomorphism in step 2.1. Thus the images of a regular system of parameters at $q$ form a cotangent basis at $p$ and, by [F6], a regular system of parameters at $p$. The local flatness criterion [F5] gives ${\mathcal{O}}_{X,p}$ flat over ${\mathcal{O}}_{G,q}$. For an arbitrary prime $p\in X$, choose a closed point $p_0$ specializing from $p$ (possible because the affine finite-type $X$ is Jacobson). The map ${\mathcal{O}}_{G,m(p)}\to {\mathcal{O}}_{X,p}$ is a localization of the flat local map at $p_0$ and remains flat. Hence $m$ is flat at every point.

$m$ is unramified. At every closed $\mathbb{C}$-point $p$ the cotangent map is an isomorphism by step 2.1, so the transitivity sequence [F3] gives ${\mathrm{\Omega }}_{X\mathrm{/}G}\otimes \kappa (p)=0$; Nakayama gives ${\mathrm{\Omega }}_{X\mathrm{/}G,p}=0$. This is a finite coherent module because $m$ is of finite presentation; if it were nonzero anywhere, its closed support in the affine Jacobson $X$ would contain a closed point, a contradiction. Thus ${\mathrm{\Omega }}_{X\mathrm{/}G}=0$ at all points, and [F4] makes $m$ formally unramified.

$m$ is étale, hence open. The morphism $m$ is of finite type over the field $\mathbb{C}$, and a finitely generated algebra over a Noetherian ring is finitely presented, so $m$ is locally of finite presentation; it is flat by step 3.1 and unramified by step 3.2, hence étale by the in-run item thm-etale-equivalent-flat-unramified-fp; therefore $m$ is universally open by the in-run item thm-etale-morphisms-open-and-quasi-finite, so the image $\mathrm{\Omega }=m(X)$ is an open subscheme of $G$, nonempty because $m(1,1,1)=1$. This proves the openness part of assertion (i); the remaining isomorphism claim is completed after the injectivity argument below.

The morphism $m$ is an isomorphism onto $\mathrm{\Omega }$. It is flat and locally of finite presentation, and surjective onto $\mathrm{\Omega }$ by construction, and, since $G$ is affine and $\mathrm{\Omega }\subseteq G$ is open, the principal opens $D_G(f)$ contained in $\mathrm{\Omega }$ cover $\mathrm{\Omega }$. For one such $D_G(f)=\mathrm{Spec}A$, its preimage is the principal open $D_X(m^{\ast }f)=\mathrm{Spec}B$ of the affine $X$; the restricted map is flat by step 3.1 and surjective because $D_G(f)\subseteq m(X)$, hence $A\to B$ is faithfully flat by [F8]. The two ring maps $b\mapsto b\otimes 1$ and $b\mapsto 1\otimes b$ from $B$ to $B{\otimes }_{A}B$ give two $(B{\otimes }_{A}B)$-points of $X$ whose images in $G$ coincide (both are the composite $\mathrm{Spec}(B{\otimes }_{A}B)\to \mathrm{Spec}A\to \mathrm{\Omega }\subseteq G$), so by step 1.2 they are equal; that is, $b\otimes 1=1\otimes b$ for every $b\in B$. Let $C=B\mathrm{/}A$ as an $A$-module. Since $A\to B$ is faithfully flat, it is injective, and [F7] makes the natural map $C\to C{\otimes }_{A}B$, $c\mapsto c\otimes 1$, injective: otherwise the nonzero map $A\to C$ taking $1$ to a nonzero kernel element would become zero after faithful tensoring. For $b\in B$, the equality $b\otimes 1=1\otimes b$ puts the image of $b\otimes 1$ in $(B{\otimes }_{A}B)\mathrm{/}(A{\otimes }_{A}B)=C{\otimes }_{A}B$ equal to zero, because $1\otimes b$ belongs to the image of $A{\otimes }_{A}B$. Thus the class of $b$ in $C$ is zero; every $b\in B$ lies in $A$, and $A=B$. Thus $m$ is an isomorphism onto $\mathrm{\Omega }$, completing (i).

The image is $\mathrm{\Omega }=U^{-}TU=U^{-}B$ because every element of $U^{-}\times T\times U$ maps to $u^{-}tu$, and $TU=B$; it equals $B^{-}U=T{U}^{-}U$ because $T$ normalizes $U^{-}$ by [F1], so $T{U}^{-}=U^{-}T$. For assertion (ii) it remains to see that $\mathrm{\Omega }$ is dense. By [F2] the group $G$ is smooth over $\mathbb{C}$, so all its local rings are regular, hence domains by [F9]; if two distinct irreducible components of $G$ met at a point $g$, the local ring ${\mathcal{O}}_{G,g}$ would have two distinct minimal primes and would not be a domain. Hence distinct irreducible components of the Noetherian scheme $G$ are disjoint, and connectedness of $G$ forces a single component: $G$ is irreducible. Since $\mathrm{\Omega }$ is a nonempty open subset of the irreducible scheme $G$, it is dense, and (ii) follows.

For (iii), the isomorphism $m$ identifies $\mathrm{\Omega }$ with $U^{-}\times T\times U$, and $(u^{-},t,u)\mapsto (u^{-},tu)$ is an isomorphism $U^{-}\times T\times U\to U^{-}\times B$ by $B=T⋉U$ [F1]; hence $\phi :U^{-}\times B\to \mathrm{\Omega }$, $\phi (u^{-},b)=u^{-}b$, is an isomorphism of varieties. It satisfies $\phi (u^{-},b)b^{\mathrm{\prime }}=\phi (u^{-},b{b}^{\mathrm{\prime }})$, so $\phi$ is equivariant for right translation by $B$ on the second factor, and the composite $\mathrm{\Omega }\stackrel{{\phi }^{-1}}{\to }{U}^{-}\times B\to U^{-}$ is a $B$-invariant morphism with a section $u^{-}\mapsto (u^{-},1)$; a morphism out of $\mathrm{\Omega }$ that is constant on $B$-orbits therefore factors uniquely through this composite, which exhibits it as the quotient morphism for the $B$-action. So $\mathrm{\Omega }\mathrm{/}B\cong U^{-}$, and the root coordinates of [F1] give $U^{-}\cong {\mathbb{A}}^{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}$, as claimed.

The Axiom of Choice is assumed in the statement and declared as the dependency The Axiom of Choice; inside the argument it is used only through the cited suppliers: [F1] inherits it from the root-exponential and Baker-Campbell-Hausdorff constructions, the in-run items thm-etale-morphisms-open-and-quasi-finite and thm-etale-equivalent-flat-unramified-fp inherit it from the flat and unramified theory, and [F8] uses it to detect maximal ideals. After the group data and the single system of parameters in step 3.1 are fixed, no further arbitrary choice is made.