I cannot wrap my head around something that seems pretty basic.
In "A micro Lie theory for state estimation in robotics", the authors derive the structure of the Lie algebra of $SO(3)$ by taking the time derivative of the group constraint. Then, assuming a constant angular velocity $\omega$, they conclude that the solution to the ordinary differential equation $\dot{R} = R[\omega]_\times$is the exponential map:
where $[\omega]_\times$ is a skew-symmetric matrix.
However, if I derive $R(t) = \exp([\omega]_\times t)$ using the derivative of the exponential map, I don't get the same expression:
$$ \begin{align} \frac{d}{dt} R(t) &= \frac{d}{dt} \exp([\omega]_\times t) = \exp([\omega]_\times t) \; J_r([\omega]_\times t) \; [\omega]_\times \mspace{40mu} J_r(\tau) \triangleq \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^k} (ad_\tau)^k \\ &= R(t) \; J_r([\omega]_\times t) \; [\omega]_\times \\ &\neq R(t)\; [\omega]_\times \mspace{40mu} \text{when $t>0$} \end{align} $$
Where did I go wrong?