Let:
$G = SL(2;\mathbb{R}) = \{ (a,b,c,d) \in \mathbb{R}^4 \ \vert \ ad-bc=1 \}$, $V=\mathbb{R}^3$, and $F:G \to \mathfrak{g}^*$ defined by:
$$F(a,b,c,d)=(\frac{d^2}{4}, -\frac{c^2}{4}, \frac{cd}{4} ) $$
with respect to a particular basis on $\mathfrak{g}^*$ (the one that arises from the Kiling form on $\mathfrak{g}$ and a particular basis $X,Y,Z$ on $\mathfrak{g}$). This gives rise a map:
$$dF_e: T_eG = \mathfrak{g} = \mathfrak{sl}(2; \mathbb{R}) = \{ (a,b,c,-a) \in \mathbb{R}^4 \} \to T_{(0,1,0)} \mathfrak{g}^* \cong \mathfrak{g}^* $$ defined by:
$$dF_e(v)(f) = v(f \circ F), \ f:V \to \mathbb{R}, \ v \in \mathfrak{sl}(2; \mathbb{R}) $$
I would like to compute a formula or some such thing for $dF_e$, ie. given $X \in \mathfrak{sl}(2; \mathbb{R})$ what is $dF_e(X) \in \mathfrak{g}^*$ with respect to the basis above?
It has been suggested to me to use a decompostion on $G$ writing $g=kan$ (possibly the Iwasawa decomposition of $G$ but not necessarily), but I am unsure how this would be helpful. Is there a particular choice of coordinate chart near $e \in G$ that would lead to the Jacobian matrix of the differential being with respect to my basis? If it helps, the chosen basis on $\mathfrak{g}$ is:
$$X = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, Y = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$
A guiding step in the right direction would very much be appreciated, thank you.
Well, we should start by computing $$dF_e(X) = \frac d{dt}\Big|_{t=0} F(\exp tX) = \frac d{dt}\Big|_{t=0} F\left(\begin{bmatrix} 1 & t\\ 0 & 1 \end{bmatrix}\right) = \frac d{dt}\Big|_{t=0} \big(\frac14,0,0\big)=0,$$ assuming your coordinates on $SL(2;\Bbb R)$ are $\begin{bmatrix} a&b\\c&d\end{bmatrix}$.
Can you do the other two?