Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\) of arbitrary characteristic and let \(P\) be a parabolic subgroup of \(G\) with unipotent radical \(P_u\). It was conjectured independently by P. Neumann and the reviewer that \(P\) has only finitely many orbits on any closed connected abelian normal subgroup of \(P\) in \(P_u\). The second author of the present paper proved this conjecture in [\textit{G. Röhrle}, Ann. Inst. Fourier 48, 1455-1482 (1998; Zbl 0933.20034)]. The proof involves some case by case considerations. In the paper under review the authors reprove this finiteness theorem for \(k=\mathbb{C}\) using the results of \textit{D. Panyushev} [Manuscr. Math. 83, 223-237 (1994; Zbl 0822.14024); Ann. Inst. Fourier 49, 1453-1476 (1999; Zbl 0944.17013)] (the proof of the latter results is not free of some case by case considerations as well). The authors prove that for the adjoint action, any \(G\)-orbit meeting an abelian ideal of the Lie algebra \(\text{Lie} B\) of a Borel subgroup \(B\) is spherical. The finiteness theorem is obtained as a consequence of this fact. A description of the normalizer of ad-nilpotent ideals in \(\text{Lie} B\) is given and the set of maximal abelian ideals of \(\text{Lie } B\) is studied. It is also proved that for simple groups \(G\) of types \(\text{A}_r\), \(\text{C}_r\), the mapping \(\nu: I\mapsto N_G(I)\) is a bijection between the set of abelian ideals \(I\) of \(\text{Lie } B\) and the set of standard parabolic subgroups of \(G\), and that this fails for all other types (the proof is elementary).