Flag manifolds, symmetric $\text{fr}t$-triples and Einstein metrics

Let $G$ be a compact connected simple Lie group and let $M=G^{\text{bb}C}\mathrm{/}P=G\mathrm{/}K$ be a generalized flag manifold. In this article we focus on an important invariant of $G\mathrm{/}K$, the so called $\text{fr}t$-root system $R_{\text{fr}t}$, and we introduce the notion of symmetric $\text{fr}t$-triples, that is triples of $\text{fr}t$-roots $\xi ,\zeta ,\eta \in R_{\text{fr}t}$ such that $\xi +\eta +\zeta =0$. We describe their properties and we present an interesting application on the structure constants of $G\mathrm{/}K$, quantities which are straightforward related to the construction of the homogeneous Einstein metric on $G\mathrm{/}K$. Next we classify symmetric $\text{fr}t$-triples for generalized flag manifolds $G\mathrm{/}K$ with second Betti number $b_2(G\mathrm{/}K)=1$, and we treat also the case of full flag manifolds $G\mathrm{/}T$, where $T$ is a maximal torus of $G$. In the last section we construct the homogeneous Einstein equation on flag manifolds $G\mathrm{/}K$ with five isotropy summands, determined by the simple Lie group $G=\text{SO}(7)$. By solving the corresponding algebraic system we classify all $\text{SO}(7)$-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.