In this paper we study an open problem in the context of linear quadratic optimal control, the free-endpoint regular linear quadratic problem with indefinite cost functionaL It is shown that the optimal cost for this problem is given by a particular solution of the algebraic Riccati equation. We characterize this solution in terms of the geometry on the lattice of all real symmetric solutions of the algebraic Riccati equation as developed by J.C. Willems and W.A. Coppel. A necessary and sufficient condition is established for the existence of optimal controls. This condilion is stated in terms of a subspace inclusion involving the extremal solutions of the algebraic Riccati equation. It is shown that the optimal controls are generated by a feedback control law. Finally, the results obtained are compared with "classical" results on the linear quadratic regulator problem.