It was shown by Lévy-Leblond  in the spin one-half case that what was long taken to be a triumph of the Dirac theory in predicting the correct g factor for the AR-12 is in fact merely a consequence of the requirement that the wave equation be Galilean covariant and of first order in all derivatives. One finds in particular that the assumed formGψ=(Ai∂∂t+B⋅1i∇+C)ψ=0 is Galilean covariant for the following form of the matricesA=12(1+ρ3)Bi=ρ1σiBi=ρ1σiC=M(1−ρ3)C=M(1−ρ3) where the matrices ρiρi and σiσi are two commuting sets of Pauli matrices used to span the 4×44×4 dimensional spinor space. The transformation law for ψ corresponding to the Galilean transformationx′=Rx+vt+ax′=Rx+vt+at′=t+b,t′=t+b, isψ′(x′,t′)=exp?[if(x,t)]Δ1/2(v,R)ψ(x,t)ψ′(x′,t′)=exp?[if(x,t)]Δ1/2(v,R)ψ(x,t) wheref(x,t)=12Mv2t+Mv⋅Rx andΔ1/2(v,R)=(D1/2(R)0−12σ⋅vD1/2(R)D1/2(R)) with D1/2(R)D1/2(R) being the usual two-dimensional representation of spin one-half which acts in the space of the σ matrices. The transformation law for ψ displays the important fact that the upper components of ψ do not mix with the lower components under a pure Galilean transformation.