Consider elementary rotations in 3D of the form (cf. wikipedia),
$$ \begin{alignat}{1} R_x(\alpha) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\[3pt] 0 & \sin \alpha & \cos \alpha \\[3pt] \end{bmatrix} \\[6pt] R_y(\beta) &= \begin{bmatrix} \cos \beta & 0 & \sin \beta \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \beta & 0 & \cos \beta \\ \end{bmatrix} \\[6pt] R_z(\gamma) &= \begin{bmatrix} \cos \gamma & -\sin \gamma & 0 \\[3pt] \sin \gamma & \cos \gamma & 0\\[3pt] 0 & 0 & 1\\ \end{bmatrix} \end{alignat} $$
Any 3D rotation $U \in SO(3)$ can be written as a product of these matrices, $$U =\prod_{k=1}^3 R_{i_k}(\theta_k)$$
If we place no further assumption on $U$, is such a factorization unique or can one always find a different order of the matrices with different rotation angles whose product is also $U$?