Summary: The class of \(\mathcal{J}\)-lattices was originally defined in the second author's thesis and subsequently by \textit{W. E. Longstaff, J. B. Nation} and \textit{O. Panaia} [Bull. Aust. Math. Soc. 58, No. 2, 245--260 (1998; Zbl 0920.47005)]. A subspace lattice \(\mathcal{L}\) on a Banach space \(X\) which is also a \(\mathcal{J}\)-lattice is called a \(\mathcal{J}\)-subspace lattice, abbreviated JSL. It is demonstrated that every single element of \(\text{Alg}\mathcal{L}\) has rank at most one. It is also shown that \(\text{Alg}\mathcal{L}\) has the strong finite rank decomposability property. Let \(\mathcal{L}_1\) and \(\mathcal{L}_2\) be subspace lattices that are also JSL's on the Banach spaces \(X_1\) and \(X_2\), respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between \(\text{Alg}\mathcal{L}_1\) and \(\text{Alg}\mathcal{L}_2\) preserves rank. Finally, we prove that every algebraic isomorphism between \(\text{Alg}\mathcal{L}_1\) and \(\text{Alg}\mathcal{L}_2\) is quasi-spatial.