The structure of complete left symmetric algebras and that of 
simple left symmetric algebras over a solvable Lie algebra have 
been studied by many authors (cf.[K], [SEG], [B]).

In [SHI] the structure of left symmetric algebras 
with a principal idempotent was studied.

In this paper, we shall study the structure of left symmetric 
algebras with a principal idempotent in $\rm I$ (resp. a principal 
nilpotent in $\rm II$) and give some examples of simple left 
symmetric algebras over a solvable Lie algebra in $\rm III$.