Homotopy exponents of real Stiefel manifolds

Let p be an odd prime. For the real Stiefel manifold Vm+k,k=SO(m+k)/SO(m), we obtain an upper bound of its p-primary homotopy exponent by decomposing ΩVm+k,k as the product of some simpler spaces for m+k⩽(p−1)2+2.