Summary: Let \(U\) be the quantized enveloping algebra of a semi-simple Lie algebra. A theorem of A. Joseph and G. Letzter [Am. J. Math. (to appear)], asserts that the subspace of \(\text{ad }U\)-finite elements of \(U\) is the direct sum of \(\text{sub-ad }U\)-modules isomorphic to the endomorphism space of some finite dimensional modules. We give here a different proof of this result. It relies on the form introduced by M. Rosso, which enables us to transfer the problem inside the dual of \(U\).