I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like:
$B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ and $E^b$ of dimension $a$ and $b$ respectively such that $E^a \times E^b$ has a fixed orientation,
$BO_a$ as the set of $a$-dimensional subspaces of $\mathbb{R}^\infty$ (without regard to orientation),
$BO_b$ as the set of $b$-dimensional subspaces of $\mathbb{R}^\infty$,
$p$ as the map that forgets orientation.
For this sphere bundle I want to apply the Gysin sequence in order to compute the cohomology with $\mathbb{Z}_2$ coefficients of $B[O_a \times O_B]^+$. So I have
$\cdots \to H^{*-1}(BO_a \times BO_b) \stackrel{\smile e}{\to} H^{*}(BO_a \times BO_b) \stackrel{p^*}{\to} H^{*}( B[O_a \times O_b]^+) \to H^{*}(BO_a \times BO_b) \to \cdots$
where $e \in H^{1}(BO_a \times BO_b)$ is the euler class of the vector bundle $\mathbb{R} \hookrightarrow E \to BO_a \times BO_b$, that you can construct from $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ by passing to the disk bundle $D^1 \hookrightarrow M_p \stackrel{p_1}{\to} BO_a \times BO_b$ and then applying a change of fiber to $\mathbb{R}$.
I know that $H^{*-1}(BO_a \times BO_b) \cong \mathbb{Z}_2[w_1, \ldots,w_{1}',\ldots]$, where $w_i$ and $w_i'$ are the i-th Stiefel-Whitney classes of the tautological bundles $\xi_a$ and $\xi_b$ and, I know that the euler class $e$ reduces $\mod\,\mathbb{Z}_2$ to the first Whitney-Stiefel class of $\mathbb{R} \hookrightarrow E \to BO_a \times BO_b$. Nevertheless I don't know how to relate $e$ with $w_1$ and $w_1'$. Any suggestion?
Thanks in advance!