Weakly closed Lie modules of nest algebras

Let $\mathcal{T}(\mathcal{N})$ be a nest algebra of operators on Hilbert space and let $\mathcal{L}$ be a weakly closed Lie $\mathcal{T}(\mathcal{N})$-module. We construct explicitly the largest possible weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{J}(\mathcal{L})$ and a weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{K}(\mathcal{L})$ such that \[ \mathcal{J}(\mathcal{L})\subseteq \mathcal{L} \subseteq \mathcal{K}(\mathcal{L}) +\mathcal{D}_{\mathcal{K}(\mathcal{L})}, \] $[\mathcal{K}(\mathcal{L}), \mathcal{T}(\mathcal{N})]\subseteq \mathcal{L}$ and $\mathcal{D}_{\mathcal{K}(\mathcal{L})}$ is a von Neumann subalgebra of the diagonal $\mathcal{T}(\mathcal{N})\cap \mathcal{T}(\mathcal{N})^*$.