  The probability distribution of the closest neighbor and farther neighbor
spacings from a given level have been studied for interacting fermion/boson
systems with and without spin degree of freedom constructed using an embedded
GOE of one plus random two-body interactions. Our numerical results demonstrate
a very good consistency with the recently derived analytical expressions using
a $3 \times 3$ random matrix model and other related quantities by Srivastava
et. al [{\it J. Phys. A: Math. Theor.} {\bf 52} 025101 (2019)]. This
establishes conclusively that local level fluctuations generated by embedded
ensembles (EE) follow the results of classical Gaussian ensembles.
