  Richard Hain and Makoto Matsumoto constructed a category of universal mixed
elliptic motives, and described the fundamental Lie algebra of this category:
it is a semi-direct product of the fundamental Lie algebra Lie$\,\pi_1(MTM)$ of
the category of mixed Tate motives over ${\bf Z}$ with a filtered and graded
Lie algebra $u$. This Lie algebra, and in particular the subspace $u$, admits a
representation as derivations of the free Lie algebra on two generators. In
this paper we study the image $E$ of this representation of $u$, starting from
some results by Aaron Pollack, who determined all the relations in a certain
filtered quotient of $E$, and gave several examples of relations in low weights
in $E$ that are connected to period polynomials of cusp forms on $SL_2({\bf
Z})$. Pollack's examples lead to a conjecture on the existence of such
relations in all depths and all weights, that we state in this article and
prove in depth 3 in all weights. The proof follows quite naturally from
Ecalle's theory moulds, to which we give a brief introduction. We prove two
useful general theorems on moulds in the appendices.
