Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as follows?



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> There is a universal formula of the form$$w(\xi^m \otimes \eta^n) = p\_{m, n} (w\_1(\xi^m), \dots, w\_m(\xi^m), w\_1(\eta^n), \dots, w\_n(\eta^n)),$$where the polynomial $p\_{m, n}$ in $m + n$ variables can be characterized as follows. If $\sigma\_1, \dots, \sigma\_m$ are the elementary symmetric functions of indeterminates $t\_1, \dots, t\_m$, and if $\sigma\_1', \dots, \sigma\_n'$ are the elementary symmetric functions of $t\_1', \dots, t\_n'$, then$$p\_{m, n}(\sigma\_1, \dots, \sigma\_m, \sigma\_1', \dots, \sigma\_n') = \prod\_{i = 1}^n \prod\_{j = 1}^n (1 + t\_i + t\_j').$$
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Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation
The keyword is "splitting principle". The problem itself is a copy of an exercise from $\S7$ in Milnor-Stasheff, which has a nice hint.


For the detailed proof see Proposition 3.2.12 in this notes 
<http://www.analg.ulg.ac.be/jps/rec/icc.pdf>


