  The classification of homogeneous compact Einstein manifolds in dimension six
is an open problem. We consider the remaining open case, namely left-invariant
Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times
S^3$. Einstein metrics are critical points of the total scalar curvature
functional for fixed volume. The scalar curvature $S$ of a left-invariant
metric $g$ is constant and can be expressed as a rational function in the
parameters determining the metric. The critical points of $S$, subject to the
volume constraint, are given by the zero locus of a system of polynomials in
the parameters. In general, however, the determination of the zero locus is
apparently out of reach. Instead, we consider the case where the isotropy group
$K$ of $g$ in the group of motions is non-trivial. When $K\not\cong
\mathbb{Z}_2$ we prove that the Einstein metrics on $G$ are given by (up to
homothety) either the standard metric or the nearly K\"ahler metric, based on
representation-theoretic arguments and computer algebra. For the remaining case
$K\cong \mathbb{Z}_2$ we present partial results.
