  This is a sequel to our previous paper (joint with Furusho). It will give a
more natural framework for constructing elements in the Hopf algebra of framed
mixed Tate motives according to Bloch and Kriz. This framework allows us to
extend our previous results to interpret all multiple zeta values (including
the divergent ones) and the multiple polylogarithms in one variable as elements
of this Hopf algebra. It implies that the pro-unipotent completion of the
torsor of paths on projective line minus three points, is a mixed Tate motive
in the sense of Bloch-Kriz. Also It allows us to interpret the multiple
logarithm as an element of this Hopf algebra as long as the products of
consecutive arguments are not 1.
