Decomposition of complex hyperbolic isometries by two complex symmetries

Let $\mathbf{PU}(2,1)$ denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^{2}$, and the group $\mathbf{SU}(2,1)$ is a 3-fold covering of $\mathbf{PU}(2,1)$: $\mathbf{PU}(2,1)=\mathbf{SU}(2,1)/\{\omega I:\omega^{3}=1\}$. We study how to decompose a given pair of isometries $(A,B)\in \mathbf{SU}(2,1)^{2}$ under the form $A=I_{1}I_{2}$ and $B=I_{3}I_{2},$ where the $I_{k}$'s are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is $\mathbb{C}$-decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].