Using the sign function method, the Lyapunov equation \(Q+ A^TXE+ E^TXA= O\), \(X= X^T\) is being studied. The eigenvalues of the pencil \(A-\lambda E\) are assumed to be contained in the open left-half plane. Several results from earlier studies with \(E= I\) are extended to the general case. A new stopping criterion is introduced which saves computational cost/work space compared to the existing criteria for sign function iterations.