# 2-plane bundle over a surface splitting as line bundles

Let $$F$$ be a closed orientable surface and let $$\xi$$ be an orientable 2-plane bundle over $$F$$. Is it possible to have $$\xi = l_1 \oplus l_2$$ for line bundles $$l_1$$ and $$l_2$$ and not have $$\xi$$ be trivial?

The line bundles $$l_1$$ and $$l_2$$ would need to either both be orientable or both be nonorientable, since $$w_1(l_1) + w_1(l_2) = w_1(\xi) = 0$$. In the case where they are both orientable, they must both be trivial (since there is no 2-torsion in $$H^1(F;\mathbb{Z})$$ every oriented line bundle is trivial) so $$\xi$$ is trivial.

Therefore my question is equivalent to: Can I add two nonorientable line bundles on $$F$$ together to get a nontrivial orientable 2-plane bundle?

First of all, an orientable line bundle is always trivial, and $$H^1(F; \mathbb{Z})$$ never has two-torsion (it's a free abelian group).
As $$w_1(\ell_1) + w_1(\ell_2) = 0$$, we see that $$w_1(\ell_1) = w_1(\ell_2)$$ and hence $$\ell_1 \cong \ell_2$$. Suppose then that $$\xi \cong \ell\oplus\ell$$ for some real line bundle $$\ell$$. Then $$w_1(\xi) = 0$$, so $$\xi$$ is orientable, and $$w_2(\xi) = w_1(\ell)^2$$. If we can choose $$\ell$$ such that $$w_1(\ell)^2 \neq 0$$, then $$\xi$$ will be non-trivial. Such an $$\ell$$ exists on any closed orientable surface of positive genus.
For example, on $$S^1\times S^1$$, let $$\ell = \pi_1^*\gamma\otimes\pi_2^*\gamma$$ where $$\pi_i : S^1\times S^1 \to S^1$$ is projection onto the $$i^{\text{th}}$$ factor, and $$\gamma$$ is the unique non-trivial real line bundle on $$S^1$$ which is nothing but the tautological line bundle on $$\mathbb{RP}^1 = S^1$$. We have $$H^*(S^1\times S^1; \mathbb{Z}_2) \cong \mathbb{Z}_2[\alpha, \beta]/(\alpha^2, \beta^2)$$ and $$w_1(\ell) = \alpha + \beta \neq 0$$, and $$w_1(\ell)^2 = \alpha\beta \neq 0$$. Therefore $$\xi = \ell\oplus\ell$$ is an orientable non-trivial bundle.
For a higher genus surface $$\Sigma_g$$, let $$f : \Sigma_g \to \Sigma_1$$ be a degree one map. Then $$f^*\ell$$ is a line bundle of the desired form on $$\Sigma_g$$.