  The Koopman-von Neumann (KvN) mechanics is an approach that was formulated
long ago to answer the question regarding the existence of a Hilbert space
representation of classical mechanics. KvN mechanics is a non-relativistic
theory, and it is not clear how spin can be included in it, since spin is
widely regarded as a relativistic property. In Eur. Phys. J. Spec. Top. {\bf
227}, 2195 (2019) it was argued that the Spohn equation [Ann. Phys. {\bf 282},
420 (2000)] is the correct classical framework for the Koopman-von Neumann
theory corresponding to the Dirac equation. However, after L\'{e}vy-Leblond's
seminal work on this topic, it became clear that spin naturally arises also
from the Galilean invariant wave equations, without any need of relativistic
considerations. Inspired by this, we propose another possibility of including
spin in the KvN formalism: the L\'{e}vy-Leblond equation in the Koopman-von
Neumann formalism can be obtained as a null reduction of the massless Dirac
equation in the Eisenhart-Duval lift metric. To illustrate the idea, we
implement it for a one-dimensional classical system without magnetic
interactions.
