  As an application of a general left-definite spectral theory, Everitt,
Littlejohn and Wellman, in 2002, developed the left-definite theory associated
with the classical Legendre self-adjoint second-order differential operator $A$
in $L^{2}(-1,1)$ which has the Legendre polynomials $\{P_{n}%
\}_{n=0}^{\infty}$ as eigenfunctions. As a consequence, they explicitly
determined the domain $\mathcal{D}(A^{2})$ of the self-adjoint operator
$A^{2}.$ However, this domain, in their characterization, does not contain
boundary conditions. In fact, this is a general feature of the left-definite
approach developed by Littlejohn and Wellman. Yet, the square of the
second-order Legendre expression is in the limit-4 case at each end point
$x=\pm1$ in $L^{2}(-1,1)$ so $\mathcal{D}(A^{2})$ should exhibit four boundary
conditions. In this paper, we show that this domain can, in fact, be expressed
using four separated boundary conditions using the classical GKN
(Glazman-Krein-Naimark) theory. In addition, we determine a new
characterization of $\mathcal{D}(A^{2})$ that involves four \textit{non-GKN}
boundary conditions. These new boundary conditions are surprisingly simple -
and natural - and are equivalent to the boundary conditions obtained from the
GKN theory.
