Let A and B be two Heisenberg translations of $$\mathrm{PU}(2,1)$$ . In this paper, we will discuss the groups $$\langle A, \rangle $$ generated by non-commuting determine when they are free. The main result paper improves an assertion made Xie et al. (Canad. Math. Bull. 56(4) (2013) 881–889). We also extend improved for in $$\mathrm{PSp}(2,1)$$

We present a characterization of minimal cones class \(C^2\) and \(C^1\) in the first Heisenberg group \(\mathbf{H}\), with an additional set examples that are not \(C^1\).