What are the motivations and examples of P. Hall Family?

Question

I am reading J. P. Serre's Lie Algebras and Lie Groups and here is how a P. Hall Family is defined:

I failed to understand the motivations of this definition. Could anyone explain it to me? Also, some simple examples of P. Hall Families will also be greatly appreciated.

Answer 1

This is important, for example, for the theory of free Lie algebras, e.g., a P. Hall basis. As with all definitions, the motivation comes afterwards, rarely before. A good reference is the book Free Lie algebras by Christophe Reutenauer. However, also the book J.-P. Serre you are reading gives motivating results afterwards. For an example, see

Examples of Free Lie Algebra