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Write $V=k^n$ and $H={\rm GL}(V)$. Let $T(V)$ denote the tensor algebra of $V$, and let $\theta$ denote the (infinite dimensional) representation of $H$ in $T(V)$. Since the unipotent group $G\subset H$ has no nontrivial characters, it is the stabilizer of some tensor $t_0\in T(V)$; see any of the books titled "Linear Algebraic Groups". The Lie algebra ${\rm Lie}(H)$ acts in $T(V)$ via $d\theta$, and $$ {\rm Lie}(G)=\{X\in {\rm Lie}(H)\ |\ (d\theta)(X)(t_0)=0\}. $$ Now for the representation $\theta$ in $T(V)$ we have $$ \theta(\exp(X))=\exp((d\theta)(X))$$ for $X\in {\rm Lie}(H)$ (this is an assertion about formal power series in $X$). Since for $X\in {\rm Lie}(G)$ we have $(d\theta)(X)(t_0)=0$, we conclude that $\theta(\exp(X))(t_0)=t_0$, that is, $\exp(X)\in G(k)$, as required.