Ord´o~nez Cabrera and Sung (2002) have proved that under certain "moment" conditions, for triangular arrays of weighted Banach-valued random variables, a.s. convergence, convergence in L1, convergence in probability, and complete convergence to 0 are equivalent, thus giving a variant of Chung's law of large numbers. We extend their result (under slightly sharper technical conditions) to symmetric random variables on simply connected step 2-nilpotent Lie groups.