Let $S$ be a closed surface and $G$ be a reductive Lie groups. Goldman ([see here][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 $C^{\infty}(M,\mathbb{R})$ admit a Lie bracket $\{\,,\}$.

Suppose $G$ is one of the following Lie groups: $GL_n(\mathbb{R}), GL_n(\mathbb{C}), SL_n(\mathbb{R}), SL_n(\mathbb{C}).$ Given any $x\in\pi_1(S)$, we define a function $f_x:M\rightarrow \mathbb{R}$ by $f_x(\rho)=\Re(\mathrm{tr}(\rho(x))),$ where $\Re$ is the real-part of a complex number. Given $x,y\in\pi_1(S)$, Goldman gave explicit formulas for $\{f,g\}$. 

In some papers the authors consider the Lie bracket between two complex valued function on $M$ and used Goldman's formula and paper as a reference. For example [Section 4 of this][2], [Page 542 of this][3] and [this][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 in $C^\infty(M,\mathbb{C})$ 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