I need an example of vector bundles $E_1 \to B$, $E_2 \to B$ such that $E_1, E_2$ are nontrivial and $E_1 \oplus E_2 $ is trivial.There aren't much of a bundles which I can prove to be nontrivial. One of them is the nontrivial line bundle over the circle (which turns the fiber upside down). I tried to check if the sum of two copies of those bundles is trivial.
It looks to me that it isn't because the gluing map is just the matrix $\operatorname{diag}(-1, -1)$ so it still turns things upside down and therefore any section must be zero on the intersection. I am not sure of it, though.
So what is the simplest example and does the bundle over the circle work?
Remark: I don't know characteristic classes and I am not allowed to use them anyway.
Question: "So what is the simplest example and does the bundle over the circle work?"
Answer: An elementary example is the real n-sphere. If $k$ is the real number field and $f:=x_1^2+\cdots + x_n^2-1$ with $A:=k[x_1,..,x_n]/(f)$ it follows there is an exact sequence
$$ 0 \rightarrow Adf \rightarrow A\{dx_1,..,dx_n\} \rightarrow \Omega^1_{A/k}\rightarrow 0$$
and $\Omega^1_{A/k}$ is a non-trivial projective $A$-module in general. Hence
$$\Omega^1_{A/k} \oplus A \cong A^n$$
is such a sum. A similar result holds for $Hom_A(\Omega^1_{A/k},A):=T_A$.
I believe the same result holds for the real $n$-sphere $S^n$ viewed as a differentiable manifold. The tangent bundle $T_{S^n}$ (and cotangent bundle $\Omega^1_{S^n}$) is non-trivial and you get a similar result:
$$T_{S^n}\oplus \mathcal{O}_{S^n} \cong \mathcal{O}_{S^n}^n.$$
https://en.wikipedia.org/wiki/Hairy_ball_theorem