$\mathfrak{h}\subsetneq \operatorname{Lie}(\operatorname{exp}(h))$ possible?

Question

consider $G$ a Lie group and the exponential map $\operatorname{exp} : \operatorname{Lie}(G)\to G$. Now let $\mathfrak{h}$ be a Lie-subalgebra of Lie(G). I more or less understand that $\mathfrak{h}\subset \operatorname{Lie}(\operatorname{exp}(h))$ but is it possible that $\mathfrak{h}\subsetneq \operatorname{Lie}(\operatorname{exp}(h))$?