  We analyze orbital effects of an in-plane magnetic field on the spin
structure of states of a gated quantum dot based in a two-dimensional electron
gas. Starting with a $k \cdot p$ Hamiltonian, we perturbatively calculate these
effects for the conduction band of GaAs, up to the third power of the magnetic
field. We quantify several corrections to the g-tensor and reveal their
relative importance. We find that for typical parameters, the Rashba spin-orbit
term and the isotropic term, $H_{43} \propto {\bf P}^2 {\bf B} \cdot
\boldsymbol{\sigma}$, give the largest contributions in magnitude. The in-plane
anisotropy of the g-factor is, on the other hand, dominated by the Dresselhaus
spin-orbit term. At zero magnetic field, the total correction to the g-factor
is typically 5-10% of its bulk value. In strong in-plane magnetic fields, the
corrections are modified appreciably.
