  We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced
word decomposition of the Weyl group element, inspired by the Bott-Samelson
factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed
into finer parts, and we write it as a finite sum of a product of two classical
Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to
consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup
$\Gamma_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new
explicit formula, expressing the triple divisor sum function in terms of a
double Dirichlet series of exponential sums, generalizing Ramanujan's formula.
