Kloosterman sums on orthogonal groups

We study Kloosterman sums on the orthogonal groups $S{O}_{3,3}$ and $S{O}_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and $p$-adic analysis.