$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups from a Riemannian geometry viewpoint. Naturally, one of the Lie groups I use the most is $\SO(3) \subset \mathbb{R}^{3 \times 3}$ equipped with the bi-invariant metric $\langle \xi, \eta\rangle = \frac{1}{2} \operatorname{tr}(\xi^T\eta )$, where $\xi,\eta \in \mathfrak{so}(3)$ are $3 \times 3$ skew-symmetric matrices. The way I parameterize $\SO(3)$ and $\mathfrak{so}(3)$ is not important ofc. I could just as easily use unit quaternions. 

Lately in my studies, I've been interested in *commutation error*, and ways to bound it. 

I was especially curious if there existed $K > 0$ such that $$\left\|\log(R^{-1}S)\right\| \leq K \left\|\log(S) - \log(R)\right\|$$ for all $R,S \in \SO(3)$ such that $\log(S), \log(R), \log(R^{-1}S)$ exists. Here, $\log(\cdot)$ is the Lie logarithm. Under my parameterization, it coincides with the matrix logarithm. 

After generating random rotation matrices $R,S$ in a script, to my surprise I found $K=1$ *every single time*. I have yet to randomly generate or analytically derive a counterexample. Once again, I am only focusing on pairs $(R,S)$ such that each term above exists. 

If true, this is a very powerful inequality! I'm surprised to see very little discussion about online. Surely, someone must've noticed this. Any idea how I could prove this?