I have an equation:
$$\mathbf{v}^b = \mathbf{R}_a^b \mathbf{v}^a$$
Where $\mathbf{v}^a, \mathbf{v}^b \in \mathbb{R}^3$ are known and I wish to compute $\mathbf{R}_a^b \in SO(3)$
Although it is straightforward to find some rotation matrix that satisfies this equation (e.g. using Wahba's problem via an SVD), it is clearly not unique since I can rotate vectors $\mathbf{v}^a$ and $\mathbf{v}^b$ about their own axes still satisfy the relationship, i.e.
$$\mathbf{v}^b = \mathbf{R}\left( \theta_2 \mathbf{\hat{v}}^b \right) \mathbf{R}_a^b \mathbf{R}\left( \theta_1 \mathbf{\hat{v}}^a \right) \mathbf{v}^a$$
where $\mathbf{R}\left( \theta \mathbf{\hat{v}} \right)$ is a rotation matrix constructed by a rotation $\theta$ about unit vector $\mathbf{\hat{v}}$
So my question is: what properties of $\mathbf{R}_a^b$ must hold true for all solutions to the equation?