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

The length of PU(2,1) relative to special elliptic isometries with fixed parameter