  Nicolas Monod showed that the evaluation map $H^*_m(G\curvearrowright
G/P)\longrightarrow H^*_m(G)$ between the measurable cohomology of the action
of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and
the measurable cohomology of $G$ is surjective with a kernel that can be
entirely described in terms of invariants in the cohomology of the maximal
split torus $A<G$. In a recent paper the authors refine Monod's result and show
in particular that the cohomology of non-alternating cocycles on $G/P$, namely
those lying in the kernel of the alternation map, is in general not trivial and
lies in the kernel of the evaluation. In this paper we describe explicitly such
non-alternating and alternating cocycles on $G/P$ in low degrees when $G$ is
either a product of isometries of real hyperbolic spaces or
$G=\mathrm{SL}(3,\mathbb{K})$, where $\mathbb{K}$ is either the real or the
complex field. As a consequence, we deduce that the comparison map
$H^*_{m,b}(G)\rightarrow H^*_m(G)$ from the measurable bounded cohomology is
injective in degree $3$, which is new for nontrivial products of isometries of
hyperbolic spaces.
