  We calculate the neutrino mass matrix up to one loop order in the MSSM
without R-parity, including the bilinears in the mass insertion approximation.
This introduces additional diagrams usually neglected in the literature. We
systematically consider the possible new diagrams, and find a few missing from
our previous work. We provide analytic expressions for the mass matrix elements
in the neutrino flavour basis, which are independent of the $H_d - L_i$ basis
choice in the Lagrangian. We compare the contributions from different diagrams,
and make ``Lagrangian-basis-independent'' estimates of when the new diagrams
need to be included. We briefly discuss the phenomenology of a toy model of
bilinear R-parity violation.
