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Since the unipotent group $G\subset {\rm GL}(n,k)$ has no nontrivial characters, it is the stabilizer of some tensor $\theta$ on $k^n$, see any of the books titled "Linear Algebraic Groups". If $X\in {\rm Lie}(G)$, then $X$ acts on $\theta$ by multiplication by 0. It follows immediately that $\exp(X)$ acts on $\theta$ by multiplication by 1, that is, $\exp(X)\in G(k)$, as required.