The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements formula_38 themselves are linearly independent. It is therefore common—if potentially confusing—to identify the formula_33's with the generators formula_40 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 formula_1 may be an algebra of formula_42 matrices, the universal enveloping of formula_1 is does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of formula_1; 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 formula_24, formula_46 and formula_47 as formula_22 matrices, but rather as symbols with no further properties (other than the commutation relations).