Homotopy exponents of real Stiefel manifolds

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