Bruhat Decomposition
Let \(G\) be a connected reductive group over an algebraically closed field, let \(B\subseteq G\) be a Borel subgroup, and let \(T\subseteq B\) be a maximal torus. Put \(W=N_G(T)/T\). Then \[ G=\bigsqcup_{w\in W}BwB \] as a disjoint union of locally closed double cosets. Equivalently, \[ G/B=\bigsqcup_{w\in W}BwB/B \] is a cell decomposition indexed by the Weyl group.
If \(w_0\) is the longest Weyl group element, then \(Bw_0B\) is open dense in \(G\).
Proof
The rank-one subgroups attached to simple roots give representatives \(\dot{s}_\alpha\in N_G(T)\) of the simple reflections. Since the Weyl group is generated by these simple reflections, products of the corresponding rank-one decompositions express \(G\) as a union of \(B\)-double cosets indexed by \(W\).
Distinct Weyl group elements give distinct double cosets because \(B\cap N_G(T)=T\). The multiplication maps from products of root subgroups onto each \(BwB\) show that the pieces are locally closed and have dimensions controlled by the length \(\ell(w)\). The cell with maximal length is open dense, hence corresponds to the longest element \(w_0\).