The asserted cell multiplication isn't quite right as it stands. First, GL(n) does not have "strict" BN-pair structure, but SL(n) does. An obvious extra element needs to be added for GL(n). 

Second, for the strict BN-pair situation of SL(n,F) and SL(n,o), the cell multiplication rules are all generated by two cases of $BwB\cdot B\sigma B$ for general element $w$ and _generating reflection_ $\sigma$, namely: this is $Bw\sigma B$ when $\ell(w\sigma)>\ell(w)$, and is $Bw\sigma B\cup BwB$ for $\ell(w\sigma)<\ell(\sigma)$. These are "axioms" for a BN-pair, but are _provable_ from the action of SL(n) on the affine building of homothety classes of $o$-lattices.