  For a system with three identical nucleons in a single-$j$ shell, the states
can be written as the angular momentum coupling of a nucleon pair and the odd
nucleon. The overlaps between these non-orthonormal states form a matrix which
coincides with the one derived by Rowe and Rosensteel [Phys. Rev. Lett. {\bf
87}, 172501 (2001)]. The propositions they state are related to the eigenvalue
problems of the matrix and dimensions of the associated subspaces. In this
work, the propositions will be proven from the symmetric properties of the $6j$
symbols. Algebraic expressions for the dimension of the states, eigenenergies
as well as conditions for conservation of seniority can be derived from the
matrix.
