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Fix a Borel subgroup $B$ and a maximal algebraic torus $T\subset B$. Conjugacy of Borel subgroups and the equality $N_G(B)=B$ identify the flag variety with $G/B$, by $gB\mapsto {}^{g}B$.

Every simultaneous conjugacy orbit on pairs contains a pair $(B,{}^{g}B)$: first conjugate its first member to $B$. The remaining allowed conjugations are by $B$. Two such pairs are in the same orbit exactly when $g^{\prime }$ belongs to $BgB$. Indeed ${}^{g^{\prime }}B={}^{bg}B$ is equivalent to $(bg{)}^{-1}{g}^{\prime }\in N_G(B)=B$. Thus there is an explicit bijectionThe Bruhat decomposition is the disjoint partition $G={\coprod }_{w\in W}B\dot{w}B$, with $W=N_G(T)/T$. Under the displayed bijection its double coset $B\dot{w}B$ becomes the orbitThis proves that the orbits are indexed by the Weyl group. Conversely, the assertion that these particular $O(w)$ exhaust the orbits without repetitions says precisely that the $B\dot{w}B$ exhaust $G$ without repetitions. Hence the orbit description and the Bruhat decomposition are equivalent, not merely two sets of the same cardinality. Their common index is the relative position of Borel subgroups.

The relative position of Borel subgroups $B_1,B_2$ is the Weyl group index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed $B$, the second has the form $gB{g}^{-1}$ and its index is the double coset $BgB$. This identifies the diagonal action on pairs of flags with the Bruhat decomposition.