  For any field K and directed graph E, we completely describe the elements of
the Leavitt path algebra L_K(E) which lie in the commutator subspace
[L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras
L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt
path algebras have the additional (unusual) property that all their Lie ideals
are (ring-theoretic) ideals, and construct examples of such rings with various
ideal structures.
