I am reading the book "Functional Differential Geometry" by Sussman, Wisdom and Farr. And i stuck with derivation of Riemann curvature operator from explicit transport in the section 8.1. I have got the line of reasoning till the equation 8.10: how differential equations are derived, how evolution operators is defined as Taylor series for differential operators $L_{g}$, and how they are composed. The book is available from the LoC, so excuse me for not reproducing all the formulas.
Then comes the second part of the equation 8.10:
$(e^{\epsilon L_{g_w}} e^{\epsilon L_{g_v}} e^{-\epsilon L_{g_w}} e^{-\epsilon L_{g_v}} I)(s_0) = (e^{\epsilon^2[L_{g_w}, L_{g_v}]+\dots}I)(s_0)$
I do not understand, why the exponentiation is here. If i manually expand the series i get directly something like $\left((\epsilon^2[L_{g_w}, L_{g_v}]+\dots)I\right)(s_0)$ on the right hand side. Am i wrong? What i do not get here?
Then from Wikipedia article on Riemann Curvature Tensor i've got why we interested only in second order term: other will just vanish after taking second derivative at $\epsilon = 0$. So the first term of Riemann operator is more or less clear to me. But where the second term $-\nabla_{[v,w]}$ does come from?
I have tried to read other books, but i found only explanations in coordinate terms, so there is no $-\nabla_{[v,w]}$ also. Then i have tried to take the derivative from Wikipedia article, and failed: i've got commutator of nablas, but not the nabla of the commutator.
Maybe anyone will suggest further readings or will give some directions for reasoning? I will appreciate any help. Thanks in advance!