My question is about finding the Lie algebra of a specific Lie group. Start with a Lie group $G$, with normal Lie subgroup $C \unlhd G$. Then define the following subgroup $\hat{G} \leq G \times G$: \begin{equation} \hat{G} = \{ (g_{1}, g_{2}) \in G \times G \mid g_{2}g_{1}^{-1} \in C \}. \end{equation} I want to show that the Lie algebra of $\hat{G}$ is the following \begin{equation} Lie(\hat{G}) = \{ (X,Y) \in Lie(G \times G) \mid X - Y \in Lie(C) \}. \end{equation} Note that I am using the fact that $Lie(\hat{G}) = \{ (X,Y) \in Lie(G \times G) \mid \exp(t(X,Y)) \in \hat{G} \text{ for all } t \}$. With this fact I was able to show containment in one direction; namely, if $\exp(t(X,Y)) = (\exp(tX), \exp(tY)) \in \hat{G}$ for all $t$, then $\exp(tY)\exp(-tX) \in C$ for all $t$, and so differentiating at $t = 0$ we find that $Y - X \in Lie(C)$.
I tried a few things for showing the other containment, but nothing which seemed to be in the right direction.
Thanks.
It is crucial for your case that $C$ is a normal subgroup, hence that $Lie(C)$ is an ideal in the Lie algebra. Then observe that if $X-Y \in Lie(C)$ we must also have $Lie(C) \ni [X-Y, Y] = [X, Y]$ since $Lie(C)$ is an ideal, hence specifically iterated Lie brackets of $X$ and $Y$ all lie in $Lie(C)$. Then you can use the Baker-Campbell-Hausdorff formula to write: $$ \exp(tX)\exp(-tY) = \exp(t(X-Y) + \frac{t^2}{2}[X,Y] + \dots) $$ for sufficiently small $t$. And all later terms in the dots are iterated Lie brackets of $X$ and $Y$, hence the expression in the exponential lies in $Lie(C)$.