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 L²(-1,1) having the Legendre polynomials as eigenfunctions. As a particular consequence, they explicitly determine the domain D(A²) of the self-adjoint operator A²: 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 = +1,-1 in L²(-1,1) meaning that D(A²) should exhibit four boundary conditions. In this talk, after a gentle crash course on left-definite theory and the classical Glazman-Krein-Naimark (GKN) theory, we show that D(A²) can, in fact, be expressed using four (separated) boundary conditions. In addition, we determine a new characterization of D(A²) that involves four 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.