  The Standard Model of particle physics can be deduced from a small number of
axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New
J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra
extension in the sense of Eilenberg. By doing so, they could deduce three
axioms of the NCG Standard Model (i.e. order zero, order one and massless
photon) from the single requirement that the extended algebra be associative.
However, their approach was only applied to the finite algebra and fails the
full model. By taking into account the differential graded structure of the
algebra of noncommutative differential forms, we obtain a formulation where the
same three axioms are deduced from the associativity of the extended
differential graded algebra, but which is now also compatible with the full
Standard Model. Finally, we present a Lorentzian version of the noncommutative
geometry of the Standard Model and we show that the three axioms still hold if
the four-dimensional manifold has a Lorentzian metric.
