Take the usual real special orthogonal group, represented by real matrices $R$ satisfying $R R^T = I$. It is well known this Lie group has Lie algebra formed of real antisymmetric matrices, matrices A satisfying $A + A^T = 0$ .
Given two real, antisymmetric matrices $A$ and $B$ one can formally construct the Baker-Campbell-Haussdorf-Dynkin formula which states \begin{align} e^A e^B &= e^{Z(A,B)},\\ Z(A,B) &= A + B + \frac{1}{2}[A,B] + ... \end{align} where the dots hide infinitely many nested commutators. It is easy to find examples of matrices, see for example here, on wikipedia, for which this series does not converge, but I am not aware of examples where both the matrices in question are antisymmetric, i.e. for which we are specifically asking about so(n) and SO(n).
Note that if it is relevant I only particularly care about the even dimensional special orthogonal group/algebra SO(2n), so(2n).
To be as explicit as possible I am asking for any of
- Necessary / sufficient conditions for convergence of the BCH formula for matrices in so(2n)
- An example of a pair of so(2n) matrices for which BCH does not converge
- (this is probably hard) some error bounds of the form "if I take terms up to k nested commutators in the BCH expansion the difference in norm between the matrix I compute and a true logarithm of $e^A e^B$ is less than some function of k"
Note that without loss of generality you can choose one of the matrices (say A) to be of the form \begin{align} \begin{pmatrix}0 & \lambda\\-\lambda & 0\end{pmatrix} \oplus 0, \end{align} where $0$ is a $2(n-1) \times 2(n-1)$ dimensional zero-matrix and $\oplus$ is the direct sum.
Edit: Note I have asked a related, although not identical question on mathoverflow here.