  Let $G$ be a simple linear algebraic group over an algebraically closed field
$K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module
$W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus
giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case
where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 +
\varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$
has highest weight $\varpi_2$, we determine for every unipotent element $u \in
G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As a part of this
result, we describe the conjugacy classes of unipotent elements of
$\operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$ in
$\operatorname{Sp}(V_1 \otimes V_2)$.
