  Let $k$ be a field of arbitrary characteristic, $A$ be a domain and
$K=\mathrm{frac}(A)$. Then
  (1) All exponential maps of $k^{[3]}$ are rigid, and we give a necessary and
sufficient condition for the triangularity of $\delta \in
\mathrm{EXP}(k^{[3]})$.
  (2) If $\delta \in \mathrm{EXP}(A^{[3]})$ such that
$\mathrm{rank}(\delta)=\mathrm{rank}(\delta_K)$, 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}.
