= Integration of a Lie algebra homomorphism
{title2=$d_e\Phi=\theta$}
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.
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