  Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider
its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this
short note, we prove that $TG$ admits a left-invariant naturally reductive
Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such
that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial
description of the same connection on the direct product $G\times \mathfrak{g}$
generalizes in a rather subtle way to $TS^7$, which is in many senses almost a
tangent Lie group.
