I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \oplus L_1 \oplus \dotsb \oplus L_k, ~~~ k > 0,
$$
where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.