Let $(\mathcal A,\varphi)$ be a noncommutative probability space, and let $\kappa_n:\mathcal A^n\to \mathbb C$ denote the $n$-th free cumulant associated to $\varphi$. For $\pi\in NC(4)$, let $\kappa_\pi(x_1,x_2,y_1,y_2)$ denote the multiplicative free cumulant over the blocks of $\pi$. Then, for every $x_1,x_2,y_1,y_2\in\mathcal A$,

\begin{align*} \kappa_2(x_1x_2,y_1y_2)=\sum_{\substack{\pi\in NC(4), \pi\vee \rho=1_4}}\kappa_\pi(x_1,x_2,y_1,y_2), \end{align*}

where $\rho=\{\{1,2\},\{3,4\}\}$ is the interval partition of $\{1,2,3,4\}$ and $1_4=\{\{1,2,3,4\}\}$ is the one-block partition.