  The non-exponential shape of the $\alpha$-process observed in supercooled
liquids is considered as one of the hallmarks of glassy dynamics and has thus
been under study for decades, but is still poorly understood. For a polar van
der Waals liquid, we show here - in line with a recent theory - that
dipole-dipole correlations give rise to an additional process in the dielectric
spectrum slightly slower than the $\alpha$-relaxation, which renders the
resulting combined peak narrower than observed by other experimental
techniques. This is reminiscent of the Debye process found in monohydroxy
alcohols. The additional peak can be suppressed by weakening the dipole-dipole
interaction via dilution with a nonpolar solvent.
