  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 $\mathcal{T}(\mathcal{N})\cap \mathcal{T}(\mathcal{N})^*$.
