  The structures of the enveloping semigroups of certain elementary finite- and
infinite-dimensional distal dynamical systems are given, answering open
problems posed by Namioka in 1982. The universal minimal system with
(topological) quasi-discrete spectrum is obtained from the infinite-dimensional
case. It is proved that, on one hand, a minimal system is a factor of this
universal system if and only if its enveloping semigroup has quasi-discrete
spectrum and that, on the other hand, such a factor need not have
quasi-discrete spectrum in itself. This leads to a natural generalisation of
the property of having quasi-discrete spectrum, which is named the
$\mathcal{W}$-property.
