Spectral gap for Kac's model of Boltzmann equation

We consider a random walk on Sn-1, the standard sphere of dimension n -1, generated by random rotations on randomly selected coordinate planes i, j with 1 less than or equal to i < j less than or equal to n. This dynamic was used by Marc Kac as a model for the spatially homogeneous Boltzmann equation. We prove that the spectral gap on Sn-1 is n(-1) up to a constant independent of n.