A relative basis for mixed Tate motives over the projective line minus three points

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.