  By Foissy's work, the bidendriform structure of the Word Quasisymmetric
Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual.
However, the only known explicit isomorphism due to Vargas does not respect the
bidendriform structure. This structure is entirely determined by so-called
totally primitive elements (elements such that the two half-coproducts vanish).
In this paper, we construct two bases indexed by two new combinatorial families
called red (dual side) and blue (primal side) biplane forests in bijection with
packed words. In those bases, primitive elements are indexed by biplane trees
and totally primitive elements by a certain subset of trees. We carefully
combine red and blue forests to get bicolored forests. A simple recoloring of
the edges allows us to obtain the first explicit bidendriform automorphism of
WQSym.
