Bruhat decomposition for a split reductive group

Put $X=G\mathrm{/}B$. The projective representation and separated torus weights in [F4] identify the chosen cocharacter's fixed scheme with the finite constant set $\{n_{w}B\}$. Invariant affine charts verify the local-affineness hypothesis of [F3]; $X$ is smooth geometrically connected and complete by [F4]. Thus [F3] gives attracting cells $Y(w)$. Their positive tangent spaces at $n_{w}B$ are the positive root spaces in $\mathfrak{g}\mathrm{/}\mathrm{Ad}(n_w)\mathfrak{b}$, indexed by ${\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}$, so $\dim Y(w)=n(w)$. The construction is over $k$ and does not replace the torus scheme by the possibly nondense set $T(k)$ over a finite field.

Over an algebraic closure the orbit $U{n}_{w}B\mathrm{/}B$ lies in $Y(w)$: conjugation by the cocharacter contracts $U$ to identity and fixes $n_{w}B$. This orbit is closed in the affine cell. Indeed, in its reduced affine closure a nonempty boundary would have a nonzero stable ideal. By [F5] a nonzero element lies in a finite-dimensional stable submodule of that ideal and yields a nonzero invariant function. That function is constant on the transitive orbit, hence on its reduced dense closure; the constant is nonzero because the function is nonzero, contradicting its vanishing on the boundary. Thus there is no boundary. By [F2], the orbit is isomorphic to $U^w$ and has dimension $n(w)$, equal to the irreducible affine space $Y(w)$; being closed it is all of $Y(w)$. Equality of these smooth locally closed schemes descends to $k$. Hence $X$ is the disjoint union of these $U$-orbits, independently of the Tits-system covering argument.

Since $B=UT$ and $n_w$ normalizes $T$, the inverse image of $Y(w)$ in $G$ is $B{n}_{w}B=U{n}_{w}B$. The product isomorphism $U^w\times U_w\to U$ follows by ordering the complementary root coordinates first in [F2]. The subgroup $U_w=U\cap n_{w}B{n}_w^{-1}$ is the orbit stabilizer, so moving its second factor across $n_w$ gives $U{n}_{w}B=U^w{n}_{w}B$. Pull back the $B$-torsor $G\to G\mathrm{/}B$ along the isomorphism $U^w\to Y(w)$. It has the explicit section $u\mapsto u{n}_w$, and therefore multiplication $(u,b)\mapsto u{n}_{w}b$ is an isomorphism $U^w\times B\to B{n}_{w}B$. These smooth locally closed cells are disjoint and cover $G$ because their flag cells do. Changing $n_w$ by an element of $T$ does not change the cell. This proves(a),(b).

For the regular dominant cocharacter, [F5] gives the open immersion $U^{-}\times B\to G$. Since $B=TU$ with its split torus and positive root groups, it is exactly the multiplication open immersion $U^{-}\times T\times U\to G$ with image $B^{-}B$. It is dense because $G$ is geometrically integral. The longest Weyl element sends $B$ to $B^{-}$, so $B{n}_{w_0}B=n_{w_0}(B^{-}B)$; both are open dense, but they are not asserted equal. Root combinatorics give $n(w_0)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ and no other $w$ has this length. Thus $Y(w_0)$ is the unique flag cell of maximal dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, while its group cell has dimension $\dim B+\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\dim G$. This proves(c).

The scheme isomorphisms in step 3.1 are over $k$, so they give $G(k)={⨆}_w{U}^w(k)n_{w}B(k)={⨆}_{w}B(k)n_{w}B(k)$; no inference from geometric density to arbitrary-field point generation is needed. Likewise every $k$-point of $G\mathrm{/}B$ lies in one of its $k$-defined affine cells and has a representative $u{n}_w\in G(k)$, so $G(k)$ acts transitively on $(G\mathrm{/}B)(k)$. Fixing the first flag in a pair leaves its stabilizer $B(k)$ acting on the second; hence the orbits on $(G\mathrm{/}B)(k)\times (G\mathrm{/}B)(k)$ are the $B(k)$-double cosets, indexed by $W$. The corresponding Tits data and inclusion rule are those of [F1], now with the actual point decomposition established. This proves(d).