I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant.
Maybe somebody here could help me with the notation.
Let $G = GL(N)$ $X_g = X_e g$ and $Y_g= Y_e g$ such that $(X_g)_{li}= \sum_{k=1}^{N} (X_e)_{ik} g_{kj}$ and $(Y_g)_{ij} = \sum_{k=1}^{N} (Y_e)_{ik}g_{k,j}$ Then the calculation shows that ( I only show the first and the last step)
$$([X,Y]_g)_{i,j}) = \sum_{k,l=1}^{N} \left((Y_g)_{k,l} \frac{\partial (X_g)_{i,j}}{\partial g_{k,l}}- (X_g)_{k,l} \frac{\partial (Y_g)_{i,j}}{\partial g_{k,l}} \right) = ([X_e,Y_e]g)_{i,j}$$
The thing is that I would understand this calculation, if I knew what $g$ actually was. From context, I would guess that $g$ is something like a right-translation operator, but I don't know. In especially, in this calculation that I abbreviated, it was used that $\delta_{m,k} \delta_{l,j} = \frac{\partial g_{m,j}}{\partial g_{k,l}}$ and I have no idea, where this comes from.
So first question: What is $g$? Second question: Why are there derivatives in the commutator?(cause $X,Y$ are defined without any derivatives)
and third question: Why do we have this orthogonality condition for the derivatives of $g$?