Summary: Nonlocal two-qubit quantum gates are represented by canonical decomposition or equivalently by operator-Schmidt decomposition. The former decomposition results in a geometrical representation such that all the two-qubit gates form tetrahedron within which the perfect entanglers form a polyhedron. On the other hand, it is known from the latter decomposition that the Schmidt number of nonlocal gates can be either 2 or 4. In this work, some aspects of the latter decomposition are investigated. It is shown that two gates differing by local operations possess the same set of Schmidt coefficients. Employing a geometrical method, it is established that Schmidt number 2 corresponds to controlled unitary gates. Furthermore, all the edges of tetrahedron and polyhedron are characterized using Schmidt strength, a measure of operator entanglement. It is found that one edge of the tetrahedron possesses the maximum Schmidt strength, implying that all the gates in the edge are maximally entangled.