All groups in this review are abelian and torsion-free. Some concepts like near-isomorphism, sufficient-isomorphism, mono-equivalence, main decomposition, the length of a decomposition, strongly-separable groups, quasi-separable groups, \(\tau\)-homogeneous groups, \(\tau\)-clipped groups for any type \(\tau\) and various kinds of types have been deployed to remedy the situation. A well known result in [\textit{A. Mader} and the second author, Proc. Am. Math. Soc. 146, No. 1, 93--96 (2018; Zbl 1435.20060)] states that if \(G\) is a torsion-free abelian group of finite rank, then \(G\) has a decomposition \(A \oplus H\) in which \(A\) is completely decomposable and \(H\) is clipped, that is, \(H\) has no rank one summand. In this case, \(A\) is unique up to isomorphism and \(H\) is unique up to near isomorphism. In the paper under review, the authors prove that for any torsion-free abelian group \(G\), if \(G=A \oplus H\) where \(A\) is completely decomposable and \(H\) is clipped, then \(A\) is unique up to isomorphism and the rank of \(H\) is uniquely determined. Moreover, they find classes of groups for which such a decomposition exists and provide examples where it does not exist. Several technical propositions are proved in order to get the main result.