  Let $G$ be a compact connected Lie group of dimension $m$. Once a
bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in
correspondence with $m\times m$ positive definite symmetric matrices. We
estimate the diameter and the smallest positive eigenvalue of the
Laplace-Beltrami operator associated to a left-invariant metric on $G$ in terms
of the eigenvalues of the corresponding positive definite symmetric matrix. As
a consequence, we give partial answers to a conjecture by Eldredge, Gordina and
Saloff-Coste; namely, we give large subsets $\mathcal S$ of the space of
left-invariant metrics $\mathcal M$ on $G$ such that there exists a positive
real number $C$ depending on $G$ and $\mathcal S$ such that
$\lambda_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all $g\in\mathcal S$. The
existence of the constant $C$ for $\mathcal S=\mathcal M$ is the original
conjecture.
