  Length spectra for Riemannian metrics are well studied, while sub-Riemannian
length spectra have been largely unexplored. Here we give the length spectrum
for a canonical sub-Riemannian structure attached to any compact Lie group by
restricting its Killing form to the sum of the root spaces. Surprisingly, the
shortest loops are the same in both the Riemannian and sub-Riemannian cases. We
provide specific calculations for SU(2) and SU(3).
