I have seen different people saying different things about this so I'm confused. Assuming that each element of the Quaternion vector represents a rotation along some axis, it does not make sense to be able to rotate something more than $\pm360^\circ$. Based on this intuition, we can normalize all elements of this 4-D vector to get values in the range $[-1, 1]$.
Although I might be wrong in my intuition/interpretation, some other people say that the $L_2$ Norm of the 4-D Quaternion vector must not exceed more than $1$. However, if all values of the Quaternion vector are $\pm1$ the $L_2$ Norm would be $2$. So I am confused on what is the range of valid values for the elements of a Quaternion vector. More specifically, what would be the valid range of values for the elements of the Quaternion vector if I want to limit the rotation in a 3D (XYZ) space to, say, $\pm30^\circ$ or $\pm45^\circ$ along each axis? I would appreciate if someone can clarify on this.
The reason that I need to know the valid ranges for the Quaternion vector elements is that I want to avoid using Euler angles to rotate 3D shapes. Instead, I want to be able to randomly sample valid Quaternion vectors. So I need to know what to set the maximum and minimum value of each of the elements so that I get a valid Quaternion rotation. Also, I need to set these bounds in an optimization algorithm.