  Let $X$ denote a metric Lie group diffeomorphic to $\mathbb{R}^3$ that admits
an algebraic open book decomposition. In this paper we prove that if $\Sigma$
is an immersed surface in $X$ whose left invariant Gauss map is a
diffeomorphism onto $\mathbb{S}^2$, then $\Sigma$ is an embedded sphere. As a
consequence, we deduce that any constant mean curvature sphere of index one in
$X$ is embedded.
