  Non-abelian U-duality originates from the construction of exceptional
Drinfel'd algebra (EDA), which extends the constriction of the classical
Drinfel'd double. This symmetry is a natural extension of Poisson--Lie
T-duality and is believed to be a symmetry of Type II string/M-theory or their
low-energy effective theories. In this paper, we consider non-abelian
U-dualities of 11- or 10-dimensional backgrounds starting with
$\mathrm{E}_{n(n)}$ EDA with $n\leq 6$ with vanishing trombone gauging. The
latter guarantees that all dual backgrounds satisfy the standard supergravity
equations of motion. In particular, when the duality includes a timelike
T-duality, we obtain solutions of M$^*$-theory or Type II$^*$ background
equations, as expected. Also starting with coboundary EDA's we provide examples
of generalised Yang--Baxter deformations of M-theory and Type IIB backgrounds.
The obtained results provide explicit examples when non-abelian U-duality works
well as a solution generating transformation.
