Lie ideals in triangular operator algebras

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if ${\mathfrak L}$ is a closed Lie ideal of the triangular operator algebra ${\mathcal A}$, then there exist a closed associative ideal ${\mathcal K}$ and a closed subalgebra ${\mathfrak D}_{\mathcal K}$ of the diagonal ${\mathcal A}\cap {\mathcal A}^*$ so that ${\mathcal K}\subseteq {\mathfrak L}\subseteq {\mathcal K}+ {\mathfrak D}_{\mathcal K}$.