As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-defnite theory associated with the classical Legendre self-adjoint second-order differential operator $A$ in $L^2(-1, 1)$ having the Legendre polynomials $\{P_n\}_{n=0}^\infty$ as eigenfunctions. As a particular consequence, they explicitly determine 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 $x=\pm 1$ in $L^2(-1, 1)$ meaning that $\mathcal{D}(A^2)$ 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 $\mathcal{D}(A^2)$ can, in fact, be expressed using four (separated) boundary conditions. In addition, we determine a new characterization of $\mathcal{D}(A^2)$ 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.