A semisimple flag variety is smooth and projective

First suppose $\mathrm{\Delta }=\mathrm{\varnothing }$. Then $\mathrm{\Phi }=\mathrm{\varnothing }$, so $\mathfrak{g}=\mathfrak{h}$ is abelian by the root decomposition. An abelian semisimple Lie algebra is zero, because it is its own solvable radical. Since $G$ is connected and smooth of dimension $\dim \mathfrak{g}=0$ over $\mathbb{C}$, it is the single reduced point $\mathrm{Spec}\mathbb{C}$ (a smooth zero-dimensional finite-type scheme is a finite disjoint union of points). Thus $T=B=G$, $U=1$, $\mathfrak{b}=0$, and the Plücker construction is $W_B={\bigwedge }^0(0)=\mathbb{C}$ with $v_B=1$. The orbit is ${\mathbb{P}}^0$, the orbit map and quotient are the identity of a point, and its single product chart is $\mathrm{Spec}\mathbb{C}\times B$. All claims hold, including dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=0$. For the rest of the proof assume $\mathrm{\Delta }\mathrm{\ne }\mathrm{\varnothing }$ and fix a simple root, so [F3], [F4] and the torsor-chart supplier apply.

The closed orbit. By [F4] the subset $X_B={\pi }_B(G)\subseteq \mathbb{P}(W_B)$ is nonempty, closed, irreducible, smooth, projective of dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, and $G$ acts on it by automorphisms transitively; the morphism ${\pi }_B$ is surjective onto $X_B$ by definition of $X_B$ as the image and is $G$-equivariant for the left action of $G$ on itself and on $\mathbb{P}(W_B)$.

Fibres and dimension. For $g_1,g_2\in G$ one has ${\pi }_B(g_1)={\pi }_B(g_2)$ if and only if $g_1^{-1}{g}_2\in {\mathrm{Stab}}_G([v_B])=B$ by [F3], that is $g_2\in g_{1}B$; hence the fibres of ${\pi }_B$ are exactly the right cosets $gB$, each a translate of the subgroup $B$ of dimension $\dim B$ by [F2]. The orbit-closure dimension count $\dim X_B=\dim G-\dim B=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ supplied by [F4] is the corresponding instance of this fibre computation together with $\dim G=\dim \mathfrak{h}+2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ and $\dim B=\dim \mathfrak{h}+\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ of [F1] and [F2]. This proves clause (i).

The quotient structure on $G\mathrm{/}B$. The map $gB\mapsto g[v_B]$ is a bijection $G\mathrm{/}B\to X_B$ by step 1.3. The product charts of Zariski sections of Borel and minimal-parabolic orbit maps cover $X_B$ by single $G$-translates of the open big-cell chart, and over each chart ${\pi }_B$ is the projection $V\times B\to V$ with right $B$ acting on the second factor. For every test scheme $S$, two lifts of a map $S\to X_B$ differ by a unique $B$-section after this Zariski cover, and lifts exist there; thus $X_B$ represents the fppf sheaf quotient $G\mathrm{/}B$. A $B$-invariant morphism $f:G\to Y$ is constant on the second factor of each product chart and hence descends to morphisms $V\to Y$ that agree on overlaps because ${\pi }_B$ is surjective as an fppf sheaf. They glue uniquely to a morphism $X_B\to Y$, proving the categorical quotient property and clause (ii).

Connectedness, projectivity and closedness are the corresponding clauses of [F4]: $X_B$ is a closed subvariety of the projective space $\mathbb{P}(W_B)$, hence projective; it is irreducible, hence connected; and it is nonempty because it contains $[v_B]={\pi }_B(1)$.

Conclusion. Steps 1.2, 1.3 and 2.2 prove clause (i), and step 2.1 proves clause (ii) using the proved product charts of Zariski sections of Borel and minimal-parabolic orbit maps. The Axiom of Choice is assumed in the statement and declared as the dependency The Axiom of Choice ([F6]); it is inherited through the representation-theoretic and orbit suppliers [F3] and [F4], and no further choice is made.