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I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors.

What I would like to know is angles (or real parts of the quaternions) by which I should rotate around given axis such that I would make the same rotation as original quaternion represents.

$q=q1*q2*q3$

Note: When first rotation is being made, second and third axis is being rotated also, second rotation affects third axis in the same manner.

The solution I am looking for is the one which makes the least rotations, thus minimizing rotations angles about first, second and third axis.

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