  Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system
$\kappa({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of
Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to
the arithmetic of elliptic curves by specializing the Euler system to points of
weights $(2,1,1)$. The aim of this article is showing that this Euler system
also encodes arithmetic information at points of weights $(1,1,0)$, concerning
the group of units of the associated number fields. The setting becomes
specially novel and intriguing when ${\bf g}$ and ${\bf h}$ specialize in
weight $1$ to $p$-stabilizations of eigenforms such that one is dual of
another. We encounter an exceptional zero phenomenon which forces the
specialization of $\kappa({\bf g}, {\bf h})$ at $(1,1,0)$ to vanish and we are
led to study the derivative of this class. The main result we obtain is the
proof of a conjecture of Darmon, Lauder and Rotger on iterated integrals and
another conjecture of Darmon and Rotger for Beilinson-Flach elements in the
adjoint setting. The main point of this paper is that the methods of previous
works, where the above conjectures are proved when the weight $1$ eigenforms
have CM, do not apply to our setting and new ideas are required. Here, a
factorization of $p$-adic $L$-functions is not available due to the lack of
critical points. Instead we resort to the principle of improved Euler systems
and $p$-adic $L$-functions to reduce our problems to questions which can be
resolved using Galois deformation theory. We expect this approach may be
adapted to prove other cases of the elliptic Stark conjecture and of its
generalizations that are appearing in the literature.
