  Let $G$ be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$
be a generalized flag manifold. In this article we focus on an important
invariant of $G/K$, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we
introduce the notion of symmetric $\fr{t}$-triples, that is triples of
$\fr{t}$-roots $\xi, \zeta, \eta\in R_{\fr{t}}$ such that $\xi+\eta+\zeta=0$.
We describe their properties and we present an interesting application on the
structure constants of $G/K$, quantities which are straightforward related to
the construction of the homogeneous Einstein metric on $G/K$. Next we classify
symmetric $\fr{t}$-triples for generalized flag manifolds $G/K$ with second
Betti number $b_{2}(G/K)=1$, and we treat also the case of full flag manifolds
$G/T$, where $T$ is a maximal torus of $G$. In the last section we construct
the homogeneous Einstein equation on flag manifolds $G/K$ with five isotropy
summands, determined by the simple Lie group $G=\SO(7)$. By solving the
corresponding algebraic system we classify all $\SO(7)$-invariant
(non-isometric) Einstein metrics, and these are the very first results towards
the classification of homogeneous Einstein metrics on flag manifolds with five
isotropy summands.
