Explicit form of the cochain twist decomposition elements

Question

When deforming an algebra using a cochain twist which in general may be written as $\mathcal{F}=\exp\left[\theta^{\mu\nu}(\partial_\mu\otimes \partial_\nu)\right]$, a notation that appears in the literature is $\mathcal{F}=f^\alpha\otimes f_\alpha$ (summation implied), so that $$u\star v = f^\alpha(u) f_\alpha(v)$$

Similarly the associator is written as $\Phi=\phi_1\otimes\phi_2\otimes\phi_3$

Can the $f$ and $\phi_i$ be written explicitly as some formal operators, and if so, how? It is unclear to me what operators would reproduce the cochain twist or star product when multiplied together as shown.