I have a Lie Algebra with a basis: $$ A_1 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \\ \end{pmatrix} \ A_2 = \begin{pmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \ A_3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} $$
The Lie brackets are $[A_1,A_2]$ = $A_3$ ; $[A_1,A_3] = A_1$ and $[A_3,A_2] = A_2$.
I want to compute $\log(\exp(A_1)\exp(A_2))$. I tried using the well known Baker–Campbell–Hausdorff formula until order 5 (with the terms already computed as on wikipedia) and also the Tables 1 and 2 of the paper 'An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications' from Casas and Murua published in 2009. The three terms of order 5 that I get are different.
Could someone confirm that or explain the mistakes that I made in the computations? I am trying to determine if there is a mistake in the results given in the paper. My computations are as follows:
The well-known result (from Wikipedia for instance) gives at order 5 (in terms of right-nested commutators): $$ \frac{[A_2, A_1, A_2, A_1,A_2] + [A_1, A_2, A_1,A_2,A_1]}{120} = \frac{A_1 + A_2}{120}. $$
The result using Table 1 of the paper gives at order 5: $$ \frac{[[[[A_2, A_1], A_1], A_2], A_2]}{180} - \frac{[[[A_2, A_1], A_1], [A_1, A_2]]}{120} - \frac{[[[A_2, A_1], A_2], [A_1, A_2]]}{360} = \frac{A_2}{180} - \frac{A_1}{120} - \frac{A_2}{360} = - \frac{A_1}{120} + \frac{A_2}{360}. $$
The result using Table 2 of the paper gives at order 5: $$ \frac{[A_1, [A_1, [[A_1, A_2], A_2]]]}{180} + \frac{[[A_1, A_2], [[A_1, A_2], A_2]]}{120} + \frac{[[A_1, [A_1, A_2]], [A_1, A_2]]}{360} = \frac{A_1}{360} + \frac{A_2}{120} + \frac{A_1}{180} = \frac{3 A_1}{360} + \frac{A_2}{120}. $$