Using a gauged flavor SU(3)-invariant hadronic Lagrangian, we calculate the cross sections for the strangeness-exchange reactions $YY\ensuremath{\leftrightarrow}N\ensuremath{\Xi}$ ($Y=\ensuremath{\Lambda},\ensuremath{\Sigma}$) in the Born approximation. These cross sections are then used in the relativistic Vlasov-Uehling-Uhlenbeck transport model to study $\ensuremath{\Xi}$ production in $\mathrm{Ar}+\mathrm{KCl}$ collisions at incident energy of 1.76A GeV and impact parameter $b=3.5$ fm. We find that including the contributions of hyperon-hyperon scattering channels strongly enhances the yield of $\ensuremath{\Xi}$, leading to the abundance ratio ${\ensuremath{\Xi}}^{\ensuremath{-}}/(\ensuremath{\Lambda}+{\ensuremath{\Sigma}}^{0})=3.38\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which is essentially consistent with the recently measured value of $(5.6\ifmmode\pm\else\textpm\fi{}{1.2}_{\ensuremath{-}1.7}^{+1.8})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ by the HADES Collaboration at GSI.
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