I have asked this question before and, while the accepted answer solved my problem back then, I am still interested in finding the rotation axis and angle. Let me rephrase the problem here:
I would like to find the rotation axis and angle needed to align two 3D vetor bases, $(\vec a, \vec b, \vec c)$ towards $(\vec d, \vec e, \vec f)$ (meaning that, after the rotation is applied, $(\vec a, \vec b, \vec c)$ equals $(\vec d, \vec e, \vec f)$). These bases are orthonormal (i.e. normalized vectors, forming 90-degree angles between any of them). The bases also have the same handedness (i.e. no mirroring is necessary to align one towards the other).
Now, due to these bases having the same handedness, there must exist a single rotation that aligns one towards the other (right?).
I would like to know the solution out of curiosity. There's no concern for stability/precision. I've tried solving the original question's accepted answer symbolically, then extracting the axis and angle from the matrix -- did not go so well. Maybe finding the eigenvector would help?