Uniqueness of the non-commutative divergence cocycle

We show that, for $n \geq 3$, 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra $T(A_n)$ with values in $\rvert T(A_n) \rvert \otimes \rvert T(A_n) \rvert$ are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, $\rvert T(A_n) \rvert$ 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 $\mathfrak{L_{2n}}$, and prove the uniqueness of the Enomoto-Satoh trace.