  Using non-minimal pure spinor superspace, Cederwall has constructed
BRST-invariant actions for $D=10$ super-Born-Infeld and $D=11$ supergravity
which are quartic in the superfields. But since the superfields have explicit
dependence on the non-minimal pure spinor variables, it is non-trivial to show
these actions correctly describe super-Born-Infeld and supergravity. In this
paper, we expand solutions to the equations of motion from Cederwall's actions
to leading order around the linearized solutions and show that they correctly
describe the interactions of $D=10$ super-Born-Infeld and $D=11$ supergravity.
