  The question of which separable C*-algebras have abelian central sequence
algebras was raised and studied by Phillips ([Ph88]) and Ando-Kirchberg
([AK14]). In this paper we give a complete answer to their question:
  A separable C*-algebra $A$ has abelian central sequence algebra if and only
if A satisfies Fell's condition.
  Moreover, we introduce a higher-dimensional analogue of Fell's condition and
show that it completely characterizes subhomogeneity of central sequence
algebras. In contrast, we show that any non-trivial extension by compact
operators has not only non-abelian but not even residually type I central
sequence algebra. In particular its central sequence algebra is not type I and
not residually finite-dimensional (RFD). Our techniques extensively use
properties of nilpotent elements in C*-algebras.
