and no other relations. We emphasize that the universal enveloping algebra "is not" the same as (or contained in) the algebra of formula_22 matrices. For example, the formula_22 matrix formula_24 satisfies formula_25, as is easily verified. But in the universal enveloping algebra, the element formula_26 does not satisfy formula_27—because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the Poincaré–Birkhoff–Witt theorem (discussed below) that the elements formula_28 are all linearly independent in the universal enveloping algebra.