  An SKT metric is a Hermitian metric on a complex manifold whose fundamental
2-form $\omega$ satisfies $\de\debar\omega=0$. Streets and Tian introduced in
\cite{sttiPlur} a Ricci-type flow that preserves the SKT condition. This flow
uses the Ricci form associated to the Bismut connection, the unique Hermitian
connection with totally skew-symmetric torsion, instead of the Levi-Civita
connection. A SKT metric is static if the (1,1)-part of the Ricci form of the
Bismut connection satisfies $\riccib=\lambda\omega$ for some real constant
$\lambda$. We study invariant static metrics on simply connected Lie groups,
providing in particular a classification in dimension 4 and constructing new
examples, both compact and non-compact, of static metrics in any dimension.
