The standard practice when doing a recursive construction seems to be to list all of the desired properties for the construction first:
We construct $(x_n)$, $(y_n)$, $(z_n)$ such that
(1)
(2)
(3)
... (a bunch of properties about the individual items $x_n,z_n,y_n$ and thei...1
In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt.
The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups.
== Statement of the theorem ==
Recall that any vector space V over a field has a basis...
