  We investigate the properties of principal elements of Frobenius Lie
algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that
any Lie algebra with a left symmetric algebra structure can be embedded, in a
natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of
Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on
simple Lie algebras, applied to the particular case of sl(m, K) alone, paves
the way to the complete classification of Frobenius and more generally
quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the
property that every derivation is an inner derivation, then every principal
element is semisimple, at least for K=C.
  As an important case, we prove that in the Lie algebra of the group of affine
motions of the Euclidean space of finite dimension, every derivation is inner.
We also bring a class of examples of Frobenius Lie algebras, that hence are
subalgebras of sl(m, K), but yet have nonsemisimple principal elements as well
as some with semisimple principal elements having nonrational eigenvalues,
where K=R or C.
