  We construct the Einstein equation for an invariant Riemannian metric on the
exceptional full flag manifold $M=G_2/T$. By computing a Gr\"obner basis for a
system of polynomials of multi-variables we prove that this manifold admits
exactly two non-K\"ahler 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-K\"ahler and not normal homogeneous Einstein metric.
