Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman ([Invariant functions on Lie groups and Hamiltonian flows of surface group representations][1]) proved that, for a fairly general class of groups $G$,  $M=Hom(\pi_1(S),G)/G$ admits a symplectic structure, where the quotient is by conjugation action. Hence the space of all "real valued" functions $C^\infty(M,\mathbb R)$ admits a Lie bracket $\{\,,\}$.

Suppose $G$ is one of the following Lie groups: $\operatorname{GL}_n(\mathbb R)$, $\operatorname{GL}_n(\mathbb C)$, $\operatorname{SL}_n(\mathbb R)$, or $\operatorname{SL}_n(\mathbb C)$. Given any $x\in\pi_1(S)$, we define a function $f_x:M\to \mathbb R$ by $f_x(\rho)=\Re(\operatorname{tr}(\rho(x)))$, where $\Re$ is the real-part of a complex number. Given $x,y\in\pi_1(S)$, Goldman gave explicit formulas for $\{f_x, f_y\}$. 

In some papers, the authors consider the Lie bracket between two complex-valued function on $M$, and use Goldman's formula and paper as a reference. For example Section 4 of [Andersen, Mattes, and Reshetikhin - The Poisson structure on the moduli space of flat connections and chord diagrams][2], Page 542 of [Bishwas and Guruprasad - Principal bundles on open surfaces and invariant functions on Lie groups][3], and [Etingof  Casimirs of the Goldman Lie algebra of a closed surface][4], considered the Lie bracket of the trace functions (not just the real  part) defined similarly as above. 

My question is: what is the Lie bracket of $C^\infty(M, \mathbb C)$, and how is it related to the Lie bracket of $C^\infty(M, \mathbb R)$?

Any kind of suggestion/reference/comment will be extremely helpful. Thanks in advance.

  [1]: http://terpconnect.umd.edu/~wmg/InvariantFunctions.pdf
  [2]: https://core.ac.uk/download/pdf/82095891.pdf
  [3]: https://www.worldscientific.com/doi/abs/10.1142/S0129167X93000285
  [4]: https://arxiv.org/abs/math/0407505