A pair of polynomials \(P = x^ n + P_{n - 1} (y)x^{n - 1} + \cdots + P_ 0(y)\), \(Q = x^ m + Q_{m - 2} (y) \cdot x^{m - 2} + \cdots + Q_ 0(y)\), \(m \geq 2\), is called a (reduced) Jacobian pair, if \(P_ xQ_ y - P_ yQ_ x = 1\). The Jacobian conjecture states, that every Jacobian pair generates \(K[x,y]\), where \(K\) is a field of characteristic zero. With a technique based on Toeplitz sequences, the author proves: If \(P,Q\) is a reduced Jacobian pair, then \(K[Q_{m - 2} (y), \dots, Q_ 0(y)] = K[y]\). With the help of a so-called fundamental system it is further shown, that the Jacobian determinant of this fundamental system is a nonzero constant and that \(K[P,Q] = K[x,y]\), if \(Q = x^ m + Q_ k(y)x^ k + Q_ r(y)x^ r\), where \(m > k > r\).