  We study a class of quasimorphisms of the free group that can be expressed as
infinite sums of Brooks quasimorphisms with some nice properties. We then
review Heuer's framework of decompositions developed in arXiv:1710.03193, and
put these quasimorphisms in that context. This allows us to prove triviality of
a number of cup products in the bounded cohomology of the free group with
trivial real coefficients. Finally, we introduce free products of
decompositions and relate them to Rolli's quasimorphisms of free products. A
technical result concerning these, which remains an open question, implies a
strong triviality result for the cup product.
