  We determine the non-abelian Born-Infeld action, including fermions, as it
results from the four-point tree-level open superstring scattering amplitudes
at order alpha'^2. We find that, after an appropriate field redefinition all
terms at this order can be written as a symmetrised trace. We confront this
action with the results that follow from kappa-symmetry and conclude that the
recently proposed non-abelian kappa-symmetry cannot be extended to cubic orders
in the Born-Infeld curvature.
