Invariance of ##SO(3)## Lie group when expressed via Euler angles

[LucaC]

TL;DR Summary

Computation of left-invariant action on $SO(3)$ expressed via Euler angles

I am trying to understand the properties of the $SO(3)$ Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions.
I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of $SO(3)$ dynamics $\mathbf{\dot{R}} = \mathbf{R}[\omega]_\times$, where $[\cdot]_\times$ is the skew-symmetric operator.

The original filter is based on the left-invariance of the dynamics equation under constant rotations $\mathbf{R}_0$, which translates into
\begin{equation}
\mathbf{\bar{R}} \doteq \mathbf{R}_0\mathbf{R} \Rightarrow \dot{\mathbf{\bar{R}}} = \mathbf{R}_0\mathbf{R}[\omega]_\times=\mathbf{\bar{R}}[\omega]_\times
\end{equation}
I have also applied the same property with the quaternion representation of $SO(3)$.

What I am trying to understand is if it is possible to apply a (non-linear) left-invariance property to the Roll-Pitch-Yaw representation of the dynamic equation, that is
\begin{cases}
\dot{\phi} = \omega_x + \text{sin}(\phi)\text{tan}(\theta)\omega_y+\text{cos}(\phi)\text{tan}(\theta)\omega_z \\
\dot{\theta} = \text{cos}(\phi)\omega_y-\text{sin}(\phi)\omega_z \\
\dot{\psi} = \frac{\text{sin}(\phi)}{\text{cos}(\theta)}\omega_y+\frac{\text{cos}(\phi)}{\text{cos}(\theta)}\omega_z \\
\end{cases}
by using some fixed rotations about xyz axes as $(\phi_0,\theta_0,\psi_0)$, in analogy to the matrix case.

My issue is that I am not able to find a transformation which defines $(\bar{\phi},\bar{\theta},\bar{\psi})$ and eventually $(\bar{\omega}_x,\bar{\omega}_y,\bar{\omega}_z)$ and satisfies the previous dynamics equation, as done previously with the rotation matrix.


Thank you all for your support