It is not difficult to see that there are non-trivial stably trivial bundles of rank $2$ for a closed surface $\Sigma$ of genus $\neq 1$ using that $T \Sigma$ is non-trivial, but what happens in the genus $1$ case? Are there stably trivial real vector bundle over $T^2$ which are non-trivial?
In case the answer is affirmative, I'm actually interested in the following situation. Suppose we have an orientable real vector bundle $E \rightarrow T^2$ of rank $2$ such that $E \oplus \epsilon^2$ is trivial, where $\epsilon^2 \rightarrow T^2$ is the trivial real vector bundle of rank $2$. Can we conclude in this case that $E$ is also trivial?
Thank you in advance.