This is done leisurely in Chapter 1 of the book, and comes across at many other places in the book. The abstract is not intended to make looking at the actual text. superfluous. It just summarizes what someone can expect to get out from reading the book.
In a nutshell:
Poisson brackets define Lie algebras. Commutators define Lie algebras.
But to appreciate how far this goes, you need to study the book, not the abstract. It will be worth your time.
The main body of the text is the whole book. Innovations on old subjects can usually not be described in a few paragraphs. Please criticise the book (or the FAQ, or the lecture quoted), not the abstract.