I'm working on trying to solve a state estimation problem in a non-matrix Lie group. I have found some good resources for state estimation in certain matrix Lie groups. For instance, in this paper by Marjanovic et al. where they aim to solve the equation
$$ dX(t) \;\; =\;\; X(t)V_0(t)dt + \sum_{i=1}^d X(t)V_i(t)dW_i(t) $$
for a matrix Lie group $G$, where $X(t)\in G$ for all $t$, the $V_i \in \mathfrak{g}$ for all $i$ (presumably linearly independent?) and $V_0$ according to them "in the most general case, is a sum of matrices from the tangent space $\mathfrak{g}$ and from the normal space."
My issue here is that the notations are evocative of matrix multiplication rather than general products in an arbitrary Lie group. I can wholeheartedly convince myself that $XV_0$ in a non-stochastic setting is the tangent vector of the flow through $X$ with $V_0 \in \mathfrak{g}$, but I'm not sure that interpretation holds up in the above equation. How exactly could one rewrite the above equation in terms of a general Lie group $G$ with Lie algebra $\mathfrak{g}$ which is independent of any particular matrix representation? For reference, in the paper they focused explicitly on $SO(n)$, but I'm in particular dealing with a Lie group that doesn't lend itself to as nice a matrix representation.