  Let $G_1$ and $G_2$ be Lie groups furnished with bi-invariant metrics and
$f:G_1\rightarrow G_2$ be a Lie group homomorphism which is also a minimal
isometric immersion. If $G_1$ is compact and connected, we prove that either
$G_1$ is isometric to a flat torus or $f$ is unstable as a harmonic map. We
also apply this result to the case in which $f$ is the inclusion of a compact,
connected Lie subgroup of a Lie group, as well as to construct several examples
of unstable harmonic maps into the orthogonal group.
