  We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding
the computation of the $\mathcal L$-invariant of the adjoint of a weight one
modular form in terms of units and $p$-units. While in our previous work with
Rotger the essential ingredient was the use of Galois deformations techniques
following the computations of Bella\"iche and Dimitrov, we propose a new
approach exclusively using the properties of Beilinson--Flach classes. One of
the key ingredients is the computation of a cyclotomic derivative of a
cohomology class in the framework of Perrin-Riou theory, which can be seen as a
counterpart to the earlier work of Loeffler, Venjakob and Zerbes. We hope that
this method could be applied to other scenarios regarding exceptional zeros,
and illustrate how this could lead to a better understanding of this setting by
conjecturally introducing a new $p$-adic $L$-function whose special values
involve information just about the unit of the adjoint (and not also the
$p$-unit), in the spirit of the conjectures of Harris and Venkatesh.
