Represent a rotation $R \in SO_3(\mathbb{R})$ by either
- intrinsic rotations $R = R_Y(\psi)R_K(\theta) R_{e_3}(\phi)$ (where ${e_3}$ is the $z$ axis, $K = R_3(\phi)e_1$ and $Y = R_K(\theta)e_3$) ;
- or extrinsic rotations $R = R_3 (\phi) R_1 (\theta) R_3(\psi)$.
I'm getting confused trying to define $e_\theta$ such that $ \partial_\theta (Ry) = e_\theta \times Ry$ for fixed $y \in \mathbb{R}^3$. When trying with intrinsic, I get $ e_\theta = R_Y (\psi) K$, and with extrinsic, I get $e_\theta = R_3 (\psi) e_1 = K$, which are different.
My reasoning is $R = AR_eC$, $\partial_\theta (Ry) = A (\partial_\theta B_e) C y = A (e \times BCy) = Ae \times ABCy = Ae \times Ry$, for $e$ the axis of rotation of $B_e$. Then apply to both forms : in one case we find $R_3(\phi) e_1$, in the other $R_Y(\psi) K$.
Where is my mistake ?