  We prove that for any associator, two specific families of coefficients of
the associator can be expressed in terms of coefficients of lower depth.
Combining these results to our notions of adjoint $p$-adic multiple zeta values
and multiple harmonic values, we obtain a new point of view on the question of
relating $p$-adic and finite multiple zeta values, and a few other application
to the study of $p$-adic multiple zeta values via explicit formulas.
