Here is my solution.

For the Ramond Ramond Sector, $${m_\rm{I}} = \frac{{\left[ {\hat a_ + ^\mu ,\hat{\tilde a}_ + ^\nu } \right]}}{2}{m_{{\rm{IIB}}}}$$

For the Neveu-Schwarz Neveu-Schwarz Sector, $${m_\rm{I}} = \frac{{\left[ {\hat d_ {-1/2} ^\mu ,\hat{\tilde d}_{-1/2} ^\nu } \right]}}{2}{m_{{\rm{IIB}}}}$$

For the Ramond Neveu-Schwarz Sector or the Neveu-Schwarz Ramond Sector, $${m_\rm{I}} = \frac{{{{\hat a}_ + }\hat d_{ - 1/2}^\mu - {{\hat{\tilde a}}_ + }\hat{ \tilde d}_{ - 1/2}^\mu }}{2}{m_{{\rm{IIB}}}}$$

My reasoning is that the mass spectrum is linear in the state vectors.