We address the removal of states with center-of-mass excitation from the SU(3) no-core shell model [SU(3)-NCSM] space. After these spurious states have been removed, the remaining center-of-mass excitation free (CMF) states form a suitable model space for describing the intrinsic dynamics of nuclei. We formulate the problem of finding the CMF states as a null-space problem for the center-of-mass harmonic oscillator lowering operator (B) over tilde ((01))(c.m) acting on the SU(3)-NCSM many-body space. We outline the construction process for the matrix representation of (B) over tilde ((01))(c.m) and provide a simple application of the method to the deuteron.