Decomposition of complex hyperbolic isometries by two complex symmetries

Let (2,1) denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space _², and the group (2,1) is a 3-fold covering of (2,1): (2,1)=(2,1)/\\\ω I:\ω³=1\\. We study how to decompose a given pair of isometries (A,B)\∈ (2,1)² under the form A=I₁I₂ and B=I₃I₂, where the Iₖ's are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is -decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].