  The most popular way to present mixing matrices of quarks (CKM) and leptons
(PMNS) is the parametrization with three mixing angles and one CP-violating
phase. There are two major options in this kind of parametrizations, one is the
original Kobayashi-Maskawa (KM) matrix, and the other is the Chau-Keung (CK)
matrix. In a new proposal by Frampton and He, a unitarity boomerang is
introduced to combine two unitarity triangles, and this new presentation
displays all four independent parameters of the KM parametrization in the quark
sector simultaneously. In this paper, we study the relations between KM and CK
parametrizations, and also consider the quark-lepton complementarity (QLC) in
the KM parametrization. The unitarity boomerang is discussed in the situation
of the CK parametrization for comparison with that in the KM parametrization in
the quark sector. Then we extend the idea of unitarity boomerang to the lepton
sector, and check the corresponding unitarity boomerangs in the two cases of
parametrizations.
