  We show that a bi-invariant metric on a compact connected Lie group $G$ is
spectrally isolated within the class of left-invariant metrics. In fact, we
prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer
$N$ such that, within a neighborhood of $g_0$ in the class of left-invariant
metrics of at most the same volume, $g_0$ is uniquely determined by the first
$N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities).
In the case where $G$ is simple, $N$ can be chosen to be two.
