= Bockstein of a Stiefel–Whitney class
{c}
{title2=$\beta(w_j(E))$}
For the Bockstein associated with $0\to\mathbb Z/2\to\mathbb Z/4\to\mathbb Z/2\to0$,
$$
\beta(w_j(E))=w_1(E)w_j(E)+(j+1)w_{j+1}(E).
$$
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.
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