A Riemannian metric is said to be Einstein if its Ricci curvature satisfies the Einstein equation , for some constant . At the present time no general existence results for Einstein metrics are known . However , there are results for many interesting justify; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt" If the Riemannian manifold is compact , then an old result of Hilbert states that is an Einstein metric if and only if is a critical point of the scalar curvature functional . For the case of compact Lie groups , the only complete work is by D’Atri and Ziller , in which they obtained a large number of left-invariant Einstein metrics that are naturally reductive . In the present work , we study existence of left-invariant Einstein metrics on several compact Lie groups , which are not naturally reductive . Then , we study the invariant Einstein metrics for all flag manifolds whose isotropy representation decompose into two inequivalent irreducible submodules . We also study the invariant Einstein metrics on some homogeneous space of some justify; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt" Moreover , we describe a 31.8pt; HEIGHT: 22.2pt" id=_x0000_i1025 type="#_x0000_t75" of a compact connected semisimple Lie group , which is consistent with a homogeneous fibration . This metrics are called adapted . We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators . Then , we describe binormal Einstein metrics which are the orthogonal sum of the normal metrics on the fiber and on the base . A natural question is to determine an Einstein adapted metric whose restriction to the fiber and projection to the base space are also Einstein metrics . This special case is also considered . As an application , we show the existence of a non-normal Einstein invariant metric on the Kowalski spaces .