H. Miller introduced a filtration of Stiefel manifolds which splits stably and proved that the starata of the filtration are vector bundles, so that Stiefel manifolds are stably equivalent to a wedge of the corresponding Thom spaces. In this paper, we shall give a finer -local stable splitting of the Stiefel manifolds by using the Adams operations.
"On a -local stable splitting of Stiefel manifolds." J. Math. Soc. Japan 54 (4) 911 - 921, October, 2002. https://doi.org/10.2969/jmsj/1191591997