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Context: The $\chi^{1}$ Fornacis cluster (Alessi 13) is one of a few openclusters of its age and distance in the Solar neighbourhood that ought tobenefit from more attention as it can serve as a cornerstone for numerousfuture studies related to star and planet formation.  Aims: We take advantage of the early installment of the third data release ofthe Gaia space mission in combination with archival data and our ownobservations, to expand the census of cluster members and revisit someproperties of the cluster.  Methods: We applied a probabilistic method to infer membership probabilitiesover a field of more than 1000 deg${^2}$ to select the most likely clustermembers and derive the distances, spatial velocities, and physical propertiesof the stars in this sample.  Results: We identify 164 high-probability cluster members (including 61 newmembers) covering the magnitude range from 5.1 to 19.6 mag in the G-band. Oursample of cluster members is complete down to 0.04 M$_{\odot}$. We derive thedistance of $108.4\pm0.3$ pc from Bayesian inference and confirm that thecluster is comoving with the Tucana-Horologium, Columba, and Carina youngstellar associations. We investigate the kinematics of the cluster from asubsample of stars with measured radial velocities and we do not detect anysignificant expansion or rotation effects in the cluster. Our results suggestthat the cluster is somewhat younger (about 30 Myr) than previously thought.Based on spectroscopic observations, we argue that the cluster ismass-segregated and that the distribution of spectral types shows littlevariation compared to other young stellar groups.  Conclusions: In this study, we deliver the most complete census of clustermembers that can be done with Gaia data alone and we use this new sample toprovide an updated picture on the 6D structure of the cluster.

$\chi^{1}$ Fornacis cluster DANCe. Census of stars, structure, and kinematics of the cluster with Gaia-EDR3