Let $S$ be a closed surface and $G$ be a reductive Lie groups. Goldman (see here) proved that for a fairly general class of groups $Gs$, $M=Hom(\pi_1(S),G)/G$ 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 $f_x(\rho)=\text{Real part of }(trace(\rho(x))).$ Given $x,y\in\pi_1(S)$, Goldman gave explicit formulas for $\{f,g\}$.

In some papers the authors considered 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, Page 542 of this and this, 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