Period obstruction to integration of a Lie algebra homomorphism ($\tilde{\Phi }(\ker p)=\{e\}$)

For a connected source, first integrate on its universal covering Lie group. The resulting map descends to the original group exactly when it kills the discrete kernel of the covering map. For the circle group, a real-linear map $t\mapsto ct$ integrates to a circle endomorphism exactly when $c\in \mathbb{Z}$, since $2\pi \mathbb{Z}$ must map into itself. This is a global period condition invisible to the zero bracket of the circle's Lie algebra.