Lectures on Symplectic Geometry by Ana Cannas da Silva (2001; revised 2006, Lecture Notes in Mathematics 1764)

Author's text, revised January 2006: math.mit.edu/classes/18.157/2017sp/lecturesonsymplecticgeometryanacannas.pdf

Cited passage: §18.3, Definition 18.7, printed p. 109: the compatible associative and Lie products defining a Poisson algebra. The preceding Definition 18.5 and Theorem 18.6 connect the geometric bracket to this algebraic structure.