Bockstein of a Stiefel–Whitney class ($\beta (w_j(E))$)

For the Bockstein associated with $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$,For a real line bundle $L$, one first has $\beta (w_1(L))=w_1(L{)}^2$. The general identity follows from the splitting principle for real vector bundles, the Bockstein derivation rule, and the elementary-symmetric-polynomial identity obtained by separating whether an index lies in a chosen $j$-element subset.