  Employing the fact that the geometry of the $N$-qubit ($N \geq 2$) Pauli
group is embodied in the structure of the symplectic polar space
$\mathcal{W}(2N-1,2)$ and using properties of the Lagrangian Grassmannian ${\rm
LGr}(N,2N)$ defined over the smallest Galois field, it is demonstrated that
there exists a bijection between the set of maximum sets of mutually commuting
elements of the $N$-qubit Pauli group and a certain subset of elements of the
$2^{N-1}$-qubit Pauli group. In order to reveal finer traits of this
correspondence, the cases $N=3$ (also addressed recently by L\'evay, Planat and
Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages,
arXiv:1305.5689]) and $N=4$ are discussed in detail. As an apt application of
our findings, we use the stratification of the ambient projective space ${\rm
PG}(2^N-1,2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where
$G \equiv {\rm SL}(2,2)\times {\rm SL}(2,2)\times\cdots\times {\rm
SL}(2,2)\rtimes S_N$, to decompose $\underline{\pi}({\rm LGr}(N,2N))$ into
non-equivalent orbits. This leads to a partition of ${\rm LGr}(N,2N)$ into
distinguished classes that can be labeled by elements of the above-mentioned
Pauli groups.
