Integration of a Lie algebra homomorphism ($d_e\Phi =\theta$)

For a connected simply connected source Lie group, every Lie algebra homomorphism integrates uniquely to a Lie group homomorphism. Along a path $\gamma$ from the identity, transport its left logarithmic velocity by $\theta$ and solve the corresponding left-invariant differential equation in the target. The Maurer-Cartan equation and preservation of brackets make the endpoint invariant under fixed-endpoint homotopies. Simple connectedness removes path dependence, and path concatenation proves multiplicativity.

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.