  The decomposition problem of the enveloping algebra of a simple Lie algebra
is reconsidered combining both the analytical and the algebraic approach,
showing its relation with the internal labelling problem with respect to a
nilpotent subalgebra. A lower bound for the number of generators of the
commutant as well as the maximal Abelian subalgebra are obtained. The
decomposition problem for the exceptional Lie algebra $G_2$ is completely
solved.
