Let $A_0$ be a closed symmetric differential operator with adjoint domain $\mathcal D(A_0^*)$. Suppose there is a finite-dimensional boundary space $E$ and a surjective boundary map \begin{align*} \Gamma=(\Gamma_1,\Gamma_2):\mathcal D(A_0^*)\to E\oplus E \end{align*} such that [integration by parts](/theorems/210) gives Green's identity \begin{align*} (A_0^*u,v)_H-(u,A_0^*v)_H=(\Gamma_1u,\Gamma_2v)_E-(\Gamma_2u,\Gamma_1v)_E \end{align*} for all $u,v\in\mathcal D(A_0^*)$. Let $L\subset E\oplus E$, and define the restriction \begin{align*} A_L:\mathcal D(A_L)\subset H\to H,\qquad \mathcal D(A_L)=\{u\in\mathcal D(A_0^*):\Gamma u\in L\},\qquad A_Lu=A_0^*u. \end{align*} Then $A_L$ is symmetric iff $L$ is isotropic for the boundary form. It is self-adjoint iff $L$ is maximal isotropic.