Editing Universal enveloping algebra (section)

===Finding a basis===
In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of <math>x_1</math> first, then factors of <math>x_2</math>, etc. For example, whenever we have a term that contains <math>x_2 x_1</math> (in the "wrong" order), we can use the relations to rewrite this as <math>x_1 x_2</math> plus a [[linear combination]] of the <math>x_j</math>'s. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order. Thus, elements of the form
:<math>x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}</math>
with the <math>k_j</math>'s being non-negative integers, span the enveloping algebra. (We allow <math>k_j=0</math>, meaning that we allow terms in which no factors of <math>x_j</math> occur.) The [[Poincaré–Birkhoff–Witt theorem]], discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.

The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements <math>x_1,\ldots, x_n</math> themselves are linearly independent. It is therefore common—if potentially confusing—to identify the <math>x_j</math>'s with the generators <math>X_j</math> of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. Although <math>\mathfrak{g}</math> may be an algebra of <math>n\times n</math> matrices, the universal enveloping of <math>\mathfrak{g}</math> does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of <math>\mathfrak{g}</math>; the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must not interpret <math>E</math>, <math>F</math> and <math>H</math> as <math>2\times 2</math> matrices, but rather as symbols with no further properties (other than the commutation relations).