  This paper is a further contribution to the extensive study by a number of
authors of the subalgebra lattice of a Lie algebra. It is shown that, in
certain circumstances, including for all solvable algebras, for all Lie
algebras over algebraically closed fields of characteristic p > 0 that have
absolute toral rank less than or equal to 1 or are restricted, and for all Lie
algebras having the one-and-a-half generation property, the conditions of
modularity and semi-modularity are equivalent, but that the same is not true
for all Lie algebras over a perfect field of characteristic three. Semi-modular
subalgebras of dimensions one and two are characterised over (perfect, in the
case of two-dimensional subalgebras) fields of characteristic different from 2,
3.
