I'm looking for two connected Lie groups $G, H$ with Lie algebras $\frak{g}, h$ and a Lie algebra homomorphism $\varphi: \frak{g} \to h$ that isn't of the form $D_0\phi$ for any Lie group homomorphism $\phi: G \to H$.
I'm thinking of looking for a locally injective $\varphi: \frak{so}(3) \to so(2)$. If I can find one then if $\varphi = D_0\phi$ for such Lie group homomorphism then we'd have:
$\phi \circ \text{exp} = \text{exp} \circ D_0\phi: \frak{}so(3) \to$ $SO(2)$. Since $\exp: \frak{so}(2) \to$ $SO(2)$ is surjective and in general exp is locally injective a known theorem states that then $\phi$ is a covering map. But this would imply that $\varphi$ is an isomorphism, which is a contradiction.
Is there such a homomorphism, or can you give a hint for another possible example?