Cartan decomposition of a two-qubit gate ($U=e^{i\chi }(A_1\otimes A_2)e^{-i(c_{x}X\otimes X+c_{y}Y\otimes Y+c_{z}Z\otimes Z)}(B_1\otimes B_2)$)

Every two-qubit gate has the displayed form with local elements of the special unitary group and real interaction parameters. The three Pauli-product generators commute, so the central exponential is the product of three interactions. Local basis changes turn an Ising coupling into either of the other Pauli-product couplings. A fixed entangling controlled phase together with arbitrary local rotations can also synthesize each variable interaction, for example through two CNOT gates enclosing a variable $z$ rotation.