In his book Classical Mechanics, Goldstein calculates the rotation matrix in z-x'-z'' convention as follows
However, I believe that his procedure is false, since the rotation matrices $B$, $C$, and $D$ mean rotating in the original coordinate system about z, x and z axes, respectively (extrinsic rotations). Instead, we should transform these matrices to new coordinate systems using similarity transformations in order to get the proper rotations about the current transformed axes z'', x' and z, respectively (intrinsic rotations).
My procedure (using his notation): $$\xi = D x$$
The rotation matrix $C$ should be transformed to the new coordinate system: $C'=D C D^{-1}$ $$\xi'=C' D x = (D C D^{-1}) D x=DCx$$
Similarily, for $B$: $B'=(DC)B(DC)^{-1}$ $$x'=B' D C x = (DC)B(DC)^{-1}(DC)x=DCBx$$
Therefore, the actual rotation matrix should be $$A=DCB$$ and not $BCD$ as Goldstein states.
I sincerely believe that my reasoning is correct, however, I know that Goldstein's book is very distinguished, and such an important error would have certainly not be missed by mistake.
What is then the correct order for multiplying matrices to obtain the rotation matrix, and why?