Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \mathfrak{g}^*$ and clearly $T^*\mathfrak{g} \cong \mathfrak{g} \times \mathfrak{g}^*$, which means that $T^*TG \cong T^*G \times T^*\mathfrak{g} \cong (G \times \mathfrak{g}^*) \times (\mathfrak{g} \times \mathfrak{g}^*)$.
If the Lie algebra $\mathfrak{g}$ can be equipped with a nondegenerate quadratic form $\langle \cdot,\cdot \rangle$ (given by the trace operator in the case of matrix Lie groups), we can use that to identify $\mathfrak{g}^* \cong \mathfrak{g}$, although we do not immediately do so.
The algebra of left-invariant smooth functions on $T^*G$ can be equipped with a Poisson bracket called the Lie-Poisson bracket, while $T^*\mathfrak{g}$ can be equipped with the canonical Poisson bracket. Thus, $T^*TG$ can be equipped with a Poisson structure which is given by the direct sum of these Poisson structures. Writing this out in coordinates $(g,\Lambda_1,X,\Lambda_2) \in (G \times \mathfrak{g}^*) \times (\mathfrak{g} \times \mathfrak{g}^*)$,
$$ \{F,G\} = \left\langle \Lambda_1, \left[ \frac{\delta F}{\delta \Lambda_1}, \frac{\delta G}{\delta \Lambda_1}\right]\right\rangle + \left\langle \frac{\delta F}{\delta X},\frac{\delta G}{\delta \Lambda_2} \right\rangle - \left\langle \frac{\delta F}{\delta \Lambda_2},\frac{\delta G}{\delta X} \right\rangle$$ for left invariant functions $F,G$ on $T^*TG$.
A couple of notes: the $g$ does not make an appearance because of left-invariance as we write down this bracket after left-trivialization (a process called left-reduction). Also, $\delta F/\delta X$ stands for the functional derivative of $F$.
This Poisson structure appears in a few sources. See Section 4.1 of https://arxiv.org/pdf/2102.10807.pdf for example.
My question is fairly elementary: I don't believe that I have come across an expression for this bracket in coordinates anywhere and I wish to see an expression of this bracket in coordinates on the Lie group $G = \mathrm{SU}(1,1)$. More specifically, the Lie algebra $\mathfrak{su}(1,1)$ of this group is given by matrices of the form $\left(\begin{matrix}it&z \\ \bar{z}&-it \end{matrix}\right)$ for $z \in \mathbb{C}$, $t \in \mathbb{R}$ and I wish to see an expression of this bracket in terms of partial derivatives of $t,z,\bar{z}$ etc. If not the full expression, I would like to know how to approach this calculation, so that I could complete it myself.
Many thanks in advance.
