  In a previous work, the author have built two families of distinguished
algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line
minus three points. The goal of this paper is to show how these cycles induce
well-defined elements in the $\HH^0$ of the bar construction of the cycle
complex and thus generated comodules over this $\HH^0$, that is a mixed Tate
motives as in Bloch and Kriz construction. In addition, it is shown that out of
the two families only ones is needed at the bar construction level. As a
consequence, the author obtains that one of the family gives a basis of the
tannakian coLie coalgebra of mixed Tate motives over $\ps$ relatively to the
tannakian coLie coalgebra of mixed Tate motives over $\Sp(\Q)$. This in turns
provides a new formula for Goncharov motivic coproduct, which arise explicitly
as the coaction dual to Ihara action by special derivations.
