Say i have a transformation matrix $T$ representing a rotation + translation inside a given base $b$. If i am given three arbitrary points $(P_1,P_2,P_3)$ and three arbitrary vectors $(u, v, w)$ respectively originating from $(P_1,P_2,P_3)$. How could i find the three chained rotations around $(u,v,w)$ that would yield the rotation applied by $T$? As the rotations are chained, only the coordinates of $P_1$ and components of $u$ are known inside $B$.
The coordinates of $P_2$ and components of $v$ are known by applying the reverse rotation to $B$. and so forth for $P_3$ and $w$
I know that in the case of Euler (and davenport for that matter) chained rotations given in an intrinsic convention, can be "converted" to an extrinsic convention simply by reversing the order. ex : rotations around $(Z,Y',X'')$ yields the same results as rotations around $(X,Y,Z)$. Does this properties hold if rotations are given around any set of vectors and origins? (ignoring the residual translation)
As of today, the only method i know of uses a numerical solver, which i would like to replace by a more subtle method if it is possible.