The simple-reflection double-coset rule and the Tits system

The independent source-cell isomorphisms of [F1] give $G(k)={⨆}_w{U}^w(k)n_{w}B(k)$, so $G(k)$ is generated by $B(k)$ and $N(k)$. This is a point statement obtained from scheme isomorphisms defined over $k$, not from algebraic generation alone. The Weyl action on Borels is free by [F3], so $B(k)\cap N(k)=T(k)$; this subgroup is normal in $N(k)$ by the normalizer definition. Thus(T1) holds. The field-point quotient description of [F3] identifies $N(k)\mathrm{/}T(k)$ with $W$, generated by its simple reflections of order two, proving(T2).

Fix a simple root $\alpha$, write $s=s_{\alpha }$ and choose $n_s$. A simple reflection permutes ${\mathrm{\Phi }}^{+}\setminus \{\alpha \}$. Order those root groups first and $U_{\alpha }$ last in the coordinates of $B$ from [F1]; conjugating their first factors by $n_s$ keeps them in $B$, while $U_{\alpha }$ becomes $U_{-\alpha }$. Consequently it suffices for(T3) to show $U_{-\alpha }(k)n_s{n}_w\subseteq B(k)n_{w}B(k)\cup B(k)n_s{n}_{w}B(k)\mathrm{.}$ If $w^{-1}\alpha$ is positive, conjugating root groups gives $U_{-\alpha }{n}_s{n}_w=n_s{n}_w{U}_{w^{-1}\alpha }$, so this set lies in the second double coset.

If $w^{-1}\alpha$ is negative, the identity element of $U_{-\alpha }$ gives the second double coset. For a nonidentity element $u$, the explicit rank-one factorization [F2] gives $u\in B_{\alpha }(k)n_s{B}_{\alpha }(k)$, where $B_{\alpha }=T{U}_{\alpha }$. Hence $u{n}_s{n}_w\in B(k)n_s{B}_{\alpha }(k)n_s{n}_w\subseteq B(k)U_{-\alpha }(k)n_w=B(k)n_w{U}_{-w^{-1}\alpha }(k)\subseteq B(k)n_{w}B(k)\mathrm{.}$ Here $n_s{B}_{\alpha }{n}_s^{-1}=T{U}_{-\alpha }$ and $n_s^2\in T$, and the final root is positive. These algebraic formulas hold over every field, including small finite fields and characteristic two. This proves(T3) in both sign cases.

The negative root parameter $u_{-\alpha }(1)$ is not in $B$: for a regular dominant cocharacter with $B=P_G(\lambda )$, its orbit parameter is $t^{-⟨\alpha ,\lambda ⟩}$ and has no limit at zero. Thus $n_{s}B{n}_s^{-1}$ is not contained in $B$, proving(T4). The four axioms yield the full Tits system with the claimed inclusion, and the displayed union with $B(k)$ at the end of the Statement is the same union with its two summands written in the opposite order. No substantive assertion is derived from that tautological equality.