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Ok, you're right. But still, if with Goldman's conventions, Goldman symplectic form is (-8) times the Weil-Petersson symplectic form (as he claims in the paper), then when you take the standard killing form on the Lie algebra, you divide Goldman symplectic form by $4$, which means that the multiplicative factor should become $-2$.
I know the reference you are referring to. The problem is that I don't really agree with what they say. They say that Goldman uses $\mathrm{trace}(XY)$ as coupling in the Lie algebra but at page 211 of "The symplectic nature of fundamental groups of surfaces", Adv. in Math. 54 (1984), no. 2, 200-225, Goldman says that he is considering the Killing form on the Lie algebra. Moreover, the second paragraph of what the authors say is very unsatisfactory.
Do you know if that result holds for non-discrete buildings as well? I’m confused about which results of that book hold in general or only in the discrete case.
