Rigidity and triangularity of an exponential map

Let $k$ be a field of arbitrary characteristic, $A$ be a domain and $K=\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}(A)$. Then (1) All exponential maps of $k^{[3]}$ are rigid, and we give a necessary and sufficient condition for the triangularity of $\delta \in \mathrm{E}\mathrm{X}\mathrm{P}(k^{[3]})$. (2) If $\delta \in \mathrm{E}\mathrm{X}\mathrm{P}(A^{[3]})$ such that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\delta )=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}({\delta }_K)$, then $\delta$ is rigid and we give a necessary and sufficient condition for the triangularity of $\delta$. When $k$ is of zero characteristic, $(1)$ is due to \cite{DD} and $(2)$ is due to \cite{KL}.