  The Standard Model of particle physics encapsulates our current best
understanding of physics at the smallest scales. A fundamental axiom of this
theory is the universality of the couplings of the different generations of
leptons to the electroweak gauge bosons. The measurement of the ratio of the
rate of decay of $W$ bosons to $\tau$-leptons and muons, $R(\tau/\mu) = B(W \to
\tau \nu_\tau)/B(W \to \mu \nu_\mu)$, constitutes an important test of this
axiom. A measurement of this quantity with a novel technique using di-leptonic
$t\bar{t}$ events is presented based on 139 fb${}^{-1}$ of data recorded with
the ATLAS detector in proton--proton collisions at $\sqrt{s}=13$ TeV. Muons
originating from $W$ bosons and those originating from an intermediate
$\tau$-lepton are distinguished using the lifetime of the $\tau$-lepton,
through the muon transverse impact parameter, and differences in the muon
transverse momentum spectra. The value of $R(\tau/\mu)$ is found to be $0.992
\pm 0.013 [\pm 0.007 (stat) \pm 0.011 (syst)]$ and is in agreement with the
hypothesis of universal lepton couplings as postulated in the Standard Model.
This is the most precise measurement of this ratio, and the only such
measurement from the Large Hadron Collider, to date.
