Commutants of reflexive algebras and classification of completely distributive subspace lattices

Let $\mathcal{L}$ be a subspace lattice on a normed space $X$ containing a nontrivial comparable element. If $T$ commutes with all the operators in $\mbox{Alg}\mathcal{L}$, then there exists a scalar $\lambda$ such that $(T-\lambda I)^2=0$. Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type $I^{(n)}$, Type $II^{(n)}$ and Type $III$, respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type $I^{(1)}$, and that nontrivial atomic Boolean subspace lattices are Type $II^{(0)}$.