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The Poisson Bracket of Length functions in the Hitchin Component

Wolpert's cosine formula on Teichmüller space gives the Weil-Petersson Poisson bracket $\{l_α, l_β\}$ for geodesic length functions $l_α,l_β$ of closed curves $α,β$ as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie. In this paper, we give a short proof of this generalization using Goldman's formula for the Poisson bracket on representation varieties of surface groups into reductive Lie groups.