  Generalizing the involution length of the complex hyperbolic plane, we obtain
that the $\alpha$-length of $\mathrm{PU}(2,1)$ is $4$, that is, every element
of $\mathrm{PU}(2,1)$ can be decomposed as the product of at most $4$ special
elliptic isometries with parameter $\alpha$. We also describe the isometries
that can be written as the product of $2$ or $3$ such special elliptic
isometries.
