  We tackle the problem of Clifford isometry compilation, i.e, how to
synthesize a Clifford isometry into an executable quantum circuit. We propose a
simple framework for synthesis that only exploits the elementary properties of
the Clifford group and one equation of the symplectic group. We highlight the
versatility of our framework by showing that several normal forms of the
literature are natural corollaries. We report an improvement of the two-qubit
depth necessary for the execution of a Clifford circuit on an LNN architecture.
We also apply our framework to the synthesis of graph states and the
codiagonalization of Pauli rotations and we improve the 2-qubit count and
2-qubit depth of circuits taken from quantum chemistry experiments.
