and ${\mathbf{L}}^N=\{{\mathbf{L}}_1,\dots,{\mathbf{L}}_N\}$, respectively. The angular velocity of the $i$th colloidal particle, $\boldsymbol{\omega}_i$, is related to its angular momentum by the equation ${\mathbf{L}}_i=\mathbb{I}_i\boldsymbol{\omega}_i$ with $\mathbb{I}_i$ the inertia tensor. In the principal-axes frames $\mathfrak{B}$, such a tensor is diagonal and constant, $\mathbb{I}_i\equiv\mathbb{I}=\text{diag}(I_{1},I_{2},I_{3})$. The transformation between the frames $\mathfrak{B}$ and $\mathfrak{S}$ is given by the rotation operator $\mathcal{R}_i$,[@jose_classical_2013] such that $\mathbf{x}=\mathcal{R}_i\,\mathbf{x}'$ where $\mathbf{x}\in\mathfrak{B}$ and $\mathbf{x}'\in\mathfrak{S}$. Hence $\boldsymbol{\omega}_i=\mathcal{R}_i\boldsymbol{\omega}_i'$, with $\boldsymbol{\omega}_i'$ the angular velocity in $\mathfrak{S}$, and $\mathbb{I}_i'=\mathcal{R}_i^\top\,\mathbb{I}\,\mathcal{R}_i$, which is neither diagonal nor constant. The angular velocity ${\boldsymbol{\omega}}_i$ can be related to the Eulerian velocities, $\dot{{\boldsymbol{\alpha}}}_i$, via the transformation ${\boldsymbol{\omega}}_i({\boldsymbol{\alpha}}_i) \equiv
\dot{{\boldsymbol{\Phi}}_i}({\boldsymbol{\alpha}}_i) = {\boldsymbol{\Lambda}}_i^\top \dot{{\boldsymbol{\alpha}}}_i$, where the dot denotes time derivative and ${\boldsymbol{\Phi}}_i$ highlights the fact that ${\boldsymbol{\omega}}_i$ is the vector accounting for the rate of change of angular displacement over the Cartesian frame $\mathfrak{B}$. The exact definition of ${\boldsymbol{\Lambda}}^\top$ can be found in Appendix \[app:projection-operator-tech\].

For simplicity, we will consider angular quantities with respect to the body-fixed frames. As such, the dynamics of colloidal particles are described by the Langevin equations[@dickinson_brownian_1985; @dickinson_brownian_1985-1] (see detailed derivation in Appendix \[app:projection-operator-tech\]) $$\begin{aligned}
 \dot{{\mathbf{r}}}_j(t)=&\,\frac{1}{m}{\mathbf{p}}_j,\quad
 \dot{{\boldsymbol{\Phi}}}_j(t)=\,\mathbb{I}^{-1}{\mathbf{L}}_j,\label{eq:1}\\
 \dot{{\mathbf{p}}}_j(t)=&\, -\frac{\partial}{\partial {\mathbf{r}}_j}V({\mathbf{r}}^N,{\boldsymbol{\alpha}}^N)-\sum_{k=1}^N\left({\boldsymbol{\Gamma}}_{jk}^{TT}{\mathbf{p}}_k+{\boldsymbol{\Gamma}}_{jk}^{TR}{\mathbf{L}}_k\right)+\sum_{k=1}^N \left( \mathbf{A}_{jk}^{TT}\mathbf{f}_k(t)+\mathbf{A}_{jk}^{TR}\mathbf{t}_k(t) \right),
 \notag
 \\
 \dot{{\mathbf{L}}}_j(t)=&
 -\frac{\partial}{\partial {\boldsymbol{\Phi}}_j}{V}({\mathbf{r}}^N,{\boldsymbol{\alpha}}^N)-(\mathbb{I}^{-1}{\mathbf{L}}_j)\times{\mathbf{L}}_j
 \,
 -\sum_{k=1}^N\left({\boldsymbol{\Gamma}}_{jk}^{RT}{\mathbf{p}}_k+{\boldsymbol{\Gamma}}_{jk}^{RR}{\mathbf{L}}_k\right)
 +\sum_{k=1}^N \left( \mathbf{A}_{jk}^{RT}\mathbf{f}_k(t)+\mathbf{A}_{jk}^{RR}\mathbf{t}_k(t) \right) \notag,\end{aligned}$$ or, more compactly, $$\begin{aligned}
 \dot{{\boldsymbol{\mathfrak{r}}}}_j(t)=&\ 