Axis of rotation of composition of rotations (Artin's Algebra) Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis $v$. Is there a geometric way of finding $v$? This is problem 4.5.10 in Artin's Algebra.
My attempts have included looking at $v$ as a differentiable function of $\theta_1$ and $\theta_2$, writing out the corresponding matrix equations, staring at a wall, and guessing. Not sure where to go from here.

A rotation around an axis is a composition of two symmetries (reflections) wrt planes which intersect at this line at half the angle. If your two rotations are represented as $R_i=S_{i1}\circ S_{i2}$ so that each plane $S_{12}$ and $S_{21}$ contains both axes, then $S_{12}=S_{21}$, and $R_1\circ R_2=S_{11}\circ S_{22}$.
