# Proving equivalence intrinsic extrinsic rotations

Rotations can be generated by skew symmetric matrices $[v]^{\times}$ as: $$R = e^{\theta[v]^{\times}}$$ Where $v$ is the normalized axis of rotation and $\theta$ the angle of rotation. Using this and the fact that $$e^{\theta[Av]^{\times}}=e^{\theta A[v]^{\times}A^T}=Ae^{\theta[v]^{\times}}A^T$$ I would like to prove the equivalence of intrinsic ($R^i$ rot. about moving axes) and extrinsic ($R^e$ rot. about static/world axes) rotations. I'm starting of as follows, but I can't quite finish it. Some help would be much appreciated. (sorry for the non-alignment)

$$R = R_3^iR_2^iR_1^i\\ =e^{\theta_3[e_3^i]^{\times}} e^{\theta_2[e_2^i]^{\times}} e^{\theta_1[e_1^i]^{\times}}\\ =e^{\theta_3[R_2^eR_1^ee_3^e]^{\times}} e^{\theta_2[R_1^ee_2^e]^{\times}} e^{\theta_1[e_1^e]^{\times}}\\ =R_2^e R_1^e R_3^e (R_2^e R_1^e)^T R_1^e R_2^e (R_1^e)^T R_1^e\\ =R_2^e R_1^e R_3^e (R_1^e)^T (R_2^e)^T R_1^e R_2^e =...\\$$

The moving frame and static frame are initially aligned. The vectors $e_1$, $e_2$ and $e_3$ are the frame axes in no particular order. I was expecting to end with something like: $$R_3^iR_2^iR_1^i = R_1^eR_2^eR_3^e$$

I can prove the following example, but the above is what I'm looking for... $$R = R_z^iR_y^iR_x^i\\ =R_z^iR_y^iR_x^e\\ =R_z^i~~R_x^eR_y^e(R_x^e)^T~~R_x^e\\ =R_z^iR_x^eR_y^e\\ =(R_x^eR_y^e)R_z^e(R_x^eR_y^e)^TR_x^eR_y^e\\ =R_x^eR_y^eR_z^e\\$$

-