  We study some Lie algebras defined by solutions to the double shuffle
equations with poles and construct families of explicit solutions to these
equations in all weights and depths. These provide universal coordinates in
which to write down `zeta elements': the images of generators of the Lie
algebra of the motivic Galois group of mixed Tate motives over the integers. We
expect that a similar statement holds for associators. In particular, these
coordinates encode algebraic relations between multiple zeta values, and enable
one to compress the currently used tables for relations between multiple zeta
values in, for example, weights $\leq 13$, already by a factor of a thousand.
The Lie algebras and groups studied here form part of a large algebraic
structure which is related to the work of Ecalle on the calculus of moulds, and
also related to the theory of universal mixed elliptic motives, and modular
forms for the full modular group.
