  We investigate the superconducting properties of Sc$_{5}$Co$_{4}$Si$_{10}$
using low-temperature resistivity, magnetization, heat capacity, and muon-spin
rotation and relaxation ($\mu$SR) measurements. We find that
Sc$_{5}$Co$_{4}$Si$_{10}$ {exhibits type-II} superconductivity with a
superconducting transition temperature $T_\mathrm{C}= 3.5 (1)$\,K. The
temperature dependence of the superfluid density obtained from transverse-field
$\mu$SR spectra is best modeled using an isotropic Bardeen-Cooper-Schrieffer
type $s$-wave gap symmetry with $2\Delta/k_\mathrm{B}T_\mathrm{C} = 2.84(2)$.
However, the zero-field muon-spin relaxation asymmetry reveals the appearance
of a spontaneous magnetic field below $T_\mathrm{C}$, indicating that
time-reversal symmetry (TRS) is broken in the superconducting state. Although
this behavior is commonly associated with non-unitary or mixed singlet-triplet
pairing, our group-theoretical analysis of the Ginzburg-Landau free energy
alongside density functional theory calculations indicates that unconventional
mechanisms are pretty unlikely. Therefore, we have hypothesized that TRS
breaking may occur via a conventional electron-phonon process.
