The genus of a finite
dimensional algebra is the difference between its dimension and the number of
elements in a minimal generating set for the algebra. In this paper the classification
of finite dimensional Lie algebras of genus one and genus two is accomplished
in four steps. First, it is shown that every such algebra is either solvable
or contains a simple subalgebra. Second, the algebras containing a simple
subalgebra are determined. Third, nonminimal genus one and two Lie algebras are
shown to have a one, two dimensional ideal, J, respectively, with genus zero
quotient. Fourth, the different possible L module structures for J are analyzed
and completely determined except for the genus two index two nilpotent
algebras.
