My question on diagonalizing the inertia tensor using Euler angles was closed as a duplicate.
The suggested duplicate is a question on converting a rotation matrix to Euler angles. My question is how to directly find the Euler angles to diagonalize the inertia tensor in terms of the entries of the inertia tensor. The inertia tensor is not a rotation matrix, so the suggested duplicate does not address my question (or parts thereof).
Part of the comments were moved to chat, but basically the only somewhat useful comment is a suggestion to diagonalize the inertia tensor by finding the eigenvectors, recover the rotation matrix from this and then convert to Euler angles. The core of my question asks if there is a method to precisely avoid this path of first diagonalizing (which presumably yields a cubic that is “hard” to solve in terms of the entries of the inertia tensor).
It is possible there isn't any simple method, and I’ll take that as an answer. I figure that, if this were possible, it would be discussed somewhere in the literature but could find anything relevant.
I suggested two possible solution paths, one of which does not work (details supplied) and the other which looks horrible. If someone has a better idea I’ll be glad to hear it.
