Uniqueness of the non-commutative divergence cocycle

We show that, for $n\ge 3$, 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra $T(A_n)$ with values in $\mid T(A_n)\mid \otimes \mid T(A_n)\mid$ are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, $\mid T(A_n)\mid$ denotes the space of cyclic words. Furthermore, we study 1-cocycles of degree zero on the Lie algebra of symplectic derivations of the free Lie algebra ${\mathfrak{L}}_{\mathfrak{2}\mathfrak{n}}$, and prove the uniqueness of the Enomoto-Satoh trace.