  It is an important problem in differential geometry to find non-naturally
reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper,
we consider this problem for some coset spaces of compact simple Lie groups. A
new method to construct invariant non-naturally reductive Einstein metrics on
normal homogeneous Einstein manifolds is presented. In particular, we show that
on the standard homogeneous Einstein manifolds, except for some special cases,
there exist plenty of such metrics. A further interesting result of this paper
is that on some compact semisimple Lie groups, there exist a large number of
left invariant non-naturally reductive Einstein metrics which are not product
metrics.
