 Integration of a Lie algebra homomorphism

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.