Generalized curvature and the equations of D=11 supergravity

It is known that, for zero fermionic sector, ψμα(x)=0, the bosonic equations of Cremmer-Julia-Scherk eleven-dimensional supergravity can be collected in a compact expression, RabαγΓbγβ=0, which is a condition on the curvature Rαβ of the generalized connection w. In this Letter we show that the equation RbcαγΓabcγβ=4i((Dˆψ)bcΓ[abc)β(ψdΓd])α, where Dˆ is the covariant derivative for the generalized connection w, collects all the bosonic equations of D=11 supergravity when the gravitino is nonvanishing, ψμα(x)≠0.