I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t) = T_e L_{g(t)}H_1 dW_t + T_e L_{g(t)}H_2 dt + T_e L_{g(t)}u(t) dt,$$ where $H_1,H_2$ are elements of the Lie algebra, $u(t)$ is a curve in the Lie algebra, and $W_t$ is a real-valued brownian motion. The SDE / stochastic integrals are in the Stratonovich sense. My knowledge of processes and SDE's on manifolds are from Elton Hsu's book. According to the construction in this book, a solution to an equation such as the above is defined by being a solution to the following euclidean SDE, for every smooth real-valued function f on G,
$$df(g(t)) = (T_e L_{g(t)}H_1)f\hspace{1mm} dW_t + (T_e L_{g(t)}H_2)f\hspace{1mm}dt + (T_e L_{g(t)}u(t))f \hspace{1mm} dt.$$
In the paper, they then consider a sort of perturbation of the curve g; $g(t)e_{\epsilon}(t)$, where e_{\epsilon}(t) is another curve in G. I don't think it is important to my question, but $e_\epsilon (t)$ is defined by a set of ODE's, equation (3.2) in the paper.
They then state that 'by Itö's formula' $g_\epsilon(t) := g(t) e_\epsilon(t)$ fulfills the SDE
$$dg_\epsilon(t) = T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}(H_1) dW_t + T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}H_2 dt + T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}u(t) dt + T_e L_{g_\epsilon(t)} (T_{e_\epsilon (t)} L_{e^{-1}_\epsilon(t)}\dot{e}_\epsilon(t) dt $$
which (my calculation) is equivalent to
$$d(g(t) e_\epsilon (t)) = T_e (L_{g(t)}\circ R_{e_\epsilon (t)})H_1\hspace{1mm} dW_t + T_e (L_{g(t)}\circ R_{e_\epsilon (t)})H_2\hspace{1mm}dt + T_e (L_{g(t)}\circ R_{e_\epsilon (t)})u(t) \hspace{1mm} dt + T_{e_\epsilon (t)} L_{g(t)} \dot{e}_\epsilon(t) dt\hspace{1mm}$$
I haven't been able to verify this, or find hints elsewhere. I have a feeling that I'm missing (or lack knowledge of) something very simple. Any hints would be greatly appreciated. Even if only for the deterministic case ($H_1 = 0$).