  A solution of the so-called fermion doubling problem in Connes'
Noncommutative Standard Model has been given by Barrett in 2006 in the form of
Majorana-Weyl conditions on the fermionic field. These conditions define a
${\cal U}_{J,\chi}$-invariant subspace of the correct physical dimension, where
${\cal U}_{J,\chi}$ is the group of Krein unitaries commuting with the
chirality and real structure. They require the KO-dimension of the total triple
to be $0$. In this paper we show that this solution is, up to some trivial
modifications, and under some mild assumptions on the finite triple, the only
one with this invariance property. We also observe that a simple modification
of the fermionic action can act as a substitute for the explicit projection on
the physical subspace.
