Let $G$ be a split group over a locally compact field $F$ with non-trivial discrete valuation. Employing the structure theory of such groups and the theory of Coxeter groups, we obtain a general formula for the decomposition of double cosets $P_1\sigma P_2$ of subgroups $P_1,P_2\subset G(F)$ containing an Iwahori subgroup into left cosets of $P_2$. When $P_1$ and $P_2$ are the same hyperspecial subgroup, we use this result to derive a formula of Iwahori for the degrees of elements of the spherical Hecke algebra.