= Solution
Fibrewise evaluation identifies
$$
L^*\otimes_\mathbb C L\cong\operatorname{End}_\mathbb C(L).
$$
The identity endomorphism is a nowhere-zero continuous section, giving the <canonical trivialization of a line bundle tensored with its dual>.
Give a complex line bundle its natural orientation as a real plane bundle. Then its <Euler class of a complex line bundle>[Euler class] equals its <First Chern class>. If $a,b$ are the standard generators of $H^2(S^2\times S^2;\mathbb Z)$, choose complex line bundles $L_1,L_2$ pulled back from the <Hopf fibration> on the two factors, with $e(L_1)=a$ and $e(L_2)=b$. The triviality of $L_i^*\otimes L_i$ gives $e(L_i^*)=-e(L_i)$. Hence, for any
$$
\alpha=pa+qb,
$$
the tensor product $L_1^{\otimes p}\otimes L_2^{\otimes q}$, with negative powers interpreted using <dual bundle>[dual bundles], has Euler class $\alpha$. Its underlying oriented real rank-two bundle is the required $E_\alpha$.
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