On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:

$\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; 4$

This fact is for instance mentioned in the comments of [this question][1], but I have been unable to find a proof of it. 

My question is: Is it true that on an 8-manifold, the analogous relation 

$\mathfrak{P}(w_4) \stackrel{?}{=} p_2 \; {\rm mod} \; 4$

holds? 


  [1]: http://mathoverflow.net/questions/61043/the-stiefel-whitney-and-pontryagin-classes-of-so3-bundles