  We study the time-inconsistent linear quadratic optimal control problem for
forward-backward stochastic differential equations with potentially indefinite
cost weighting matrices for both the state and the control variables. Our
research makes two contributions. Firstly, we introduce a novel type of Riccati
equation system with parameters and constraint conditions, known as the
generalized equilibrium Riccati equation. This equation system offers a
comprehensive solution for the closed-loop equilibrium strategy of the problem
at hand. Secondly, we establish the well-posedness of the generalized
equilibrium Riccati equation for the one-dimensional case, provided certain
conditions are met.
