I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-torsion part better.



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> Is there a reference which describes the difference between $p(E\oplus F)$ and $p(E)\cdot p(F)$, perhaps in terms of Bocksteins of Stiefel-Whitney classes of $E$ and $F$?
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This question was previously part of [Whitney sum formula for Pontryagin classes I](https://mathoverflow.net/questions/222576/whitney-sum-formula-for-pontryagin-classes); Qiaochu Yuan's answer to that question might be helpful.



Whitney sum formula for Pontryagin classes II
Brown, Edgar H., Jr.
The cohomology of BSOn and BOn with integer coefficients. 
Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288. 


Theorem 1.6, last sentence:


Under Whitney sum,
$p\_q\mapsto \sum\_j r\_{2q-j}\otimes r\_j$, where
$r\_{2s} = p\_s$ and $r\_{2s+1} = (\delta w\_{2s})^2+ p\_s\delta w\_1$.


