We give a necessary condition for the existence of a fundamental solution for a bi-invariant differential operator on the nilpotent Lie group \({\mathbb{T}}^ p\times N\), with N nilpotent simply connected. A sufficient condition is obtained using the P-convexity of \({\mathbb{T}}^ p\times N\). We also prove the existence of a fundamental solution for bi-invariant operators on a simply connected solvable Lie group.