Let $A$ be a unital associative algebra over $\mathbb{C}$ with unit $1_A$, and let $\varphi: A \to \mathbb{C}$ be a unital linear functional satisfying the trace identity $\varphi(ab)=\varphi(ba)$ for all $a,b \in A$. Let $n \in \mathbb{N}$ and let $x_1,\dots,x_n \in A$. Then for every integer $k$ with $0 \leq k < n$,

\begin{align*} \varphi(x_1\cdots x_n)=\varphi(x_{k+1}\cdots x_nx_1\cdots x_k). \end{align*}

Here the empty product is interpreted as the unit $1_A$.

Consequently, for any integer $r \in \mathbb{N}$ and any tuple $x=(x_1,\dots,x_r) \in A^r$, define the joint law $L_x: \mathbb{C}\langle Z_1,\dots,Z_r\rangle \to \mathbb{C}$ by $L_x(P)=\varphi(P(x_1,\dots,x_r))$. If two words in the alphabet $\{Z_1,\dots,Z_r\}$ differ by a cyclic rotation, then their $L_x$-moments are equal. Hence the tracial word-moment data are determined by choosing one representative from each cyclic equivalence class of words.