Stiefel orientations: existence and constructions

This article is devoted to a certain class of second order structures on vector bundles, so called Stiefel orientations. Their relation to Stiefel–Whitney classes and to each other is investigated. The question of their existence turns out to be closely related to some calculations in the classifying spaces. The main (but not the most difficult) constructive result is the following:
THEOREM 3: If $z_k$ is а $k$-dimensional Stiefel orientation on a bundle $\xi^n$, $1\leqslant k<m\leqslant n$, and $\binom{k}{i}$ is odd for all $i=1,2,\dots,m-k$, then (1) there exists a unique expansion
$$ Sq^{m-k}z_k=\pi^*(y_m)+\sum_{i=k}^{m-1}\pi^*(y_{m-i})Sq^{i-k}z_k, $$
$\pi: E(V_{n-k}(\xi))\mapsto E(V_{n-m}(\xi))$ being the standard projection and $\dim y_{m-i}=m-i$; (2) the class $y_m$ is an $m$-dimensional Stiefel orientation on $\xi^n$.