I've understand that quaternions do not have handness but rotation matricies derived from unit quaternions does.
The following formula is given by wikipedia for quaternion to rotation matrix conversion :
Given the unit quaternion $q = w + xi + yj + zk$ , the equivalent left-handed (Post-Multiplied) 3×3 rotation matrix is $$ Q = \begin{bmatrix} 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\ 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\ 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2 \end{bmatrix} . $$
As mentioned, this formula is relative to a left-handed coordinate frame. What's the right-handed counterpart ?
Best : ) ,
Hint:
Given the unit quaternion $q = w + x \vec i + y\vec j + z \vec k=w+ \vec v$, the matrix $Q$ results from the representation of a rotation of a vector $\vec p=p_x \vec i+p_y \vec j+ p_z \vec k$ as: $$ Q(\vec p)= q \vec p q^{-1} $$ The resulting rotation is a rotation of angle $\theta$ around an axis oriented by a versor $\vec u$ such that:
$$ \cos \frac{\theta}{2}=\frac{w}{|q|} \qquad \sin \frac{\theta}{2}=\frac{|\vec v|}{|q|} $$ and $$ \vec u=\frac{\vec v}{|\vec v|} $$
This is a counter-clockwise rotation $R_{\vec u, \theta}$ around the axis $\vec u$.
You can find the clockwise rotation around the same axis changing the angle to $-\theta$, or inverting the orientation of the versor $\vec u$ (note that if you perform all the two transformation the rotation remain the same).
What does this means for the quaternion $q$ ?