  We show how the Weil pairing can be used to evaluate the assigned characters
of an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class
$[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given
$\mathcal{O}$-oriented elliptic curves $(E, \iota)$ and $(E', \iota') =
[\mathfrak{a}](E, \iota)$. When specialized to ordinary elliptic curves over
finite fields, our method is conceptually simpler and often somewhat faster
than a recent approach due to Castryck, Sot\'akov\'a and Vercauteren, who rely
on the Tate pairing instead. The main implication of our work is that it breaks
the decisional Diffie-Hellman problem for practically all oriented elliptic
curves that are acted upon by an even-order class group. It can also be used to
better handle the worst cases in Wesolowski's recent reduction from the
vectorization problem for oriented elliptic curves to the endomorphism ring
problem, leading to a method that always works in sub-exponential time.
