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We show that a bi-invariant metric on a compact connected Lie group $G$ isspectrally isolated within the class of left-invariant metrics. In fact, weprove 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-invariantmetrics 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.

Spectral isolation of bi-invariant metrics on compact Lie groups