@SpencerKraisler: I realized my error, which was applying the cosine function to reverse the inequality on a range where cosine is not strictly decreasing. That's why the cosine'd inequality fails but it doesn't give a counterexample to your inequality. Things go better for the quaternions, and you'd think that would also work for $\mathrm{SO}(3)$, but I'm checking the details.

If you remove the $\mathbb{RP}^2$ of elements in $\mathrm{SO}(3)$ with trace equal to the mimimum value of $-1$, there is a well-defined smooth logarithm on the open ball that remains. It satisfies $\exp(\log A) = A$. For unit quaternions (or, equivalently, $\mathrm{SU}(2)$), there is a well-defined, smooth logarithm after you remove the single element $-1$ (equivalently, $-I_2$).

@RamiroLafuente: Well, in fact, it does hold, in the sense that every compact group $G$ acting effecively on $S^4$ is a subgroup of $\mathrm{SO}(5)$, but I don't know any general result that would imply that, if $g_t$ is a continuous $1$-parameter family of Riemannian metrics on a closed manifold $M$, then every $t_0$ has an open neighborhood $U$ such that the isometry groups of the $g_t$ for $t\in U$ are subgroups of the isometry group of $g_{t_0}$.

@RamiroLafuente: It's not clear that there isn't some Riemannian metric on $S^4$ whose isometry group has dimension $8$ but is not isomorphic to a subgroup of $\mathrm{SO}(5)$.