Homogeneous Einstein metrics on G_2/T

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-Kähler and not normal homogeneous Einstein metric.

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