I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decomposable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)