In this paper, we are interested in global adaptive stabilization of nonlinear systems where we know a linearly parametrized stabilizer and a Lyapunov function for the ideal closed-loop system such that the standard estimator is implementable, but this Lyapunov function is not strict, i.e., its derivative is only negative semi-definite. Our motivation to consider this situation stems from the fact that, for many physical systems, the natural Lyapunov function candidate is the total energy, which is always not strict. To complete the stability proof in this case it is necessary to add a-rather restrictive-detectability assumption. Our main contribution is to show that it is possible to overcome this obstacle by adding to the parameter estimator an identification error coming from an indirect identifier. We thus replace the detectability assumption by a new condition that essentially requires that the negative-definite terms appearing in the Lyapunov function derivative dominate, the uncertain terms that cannot be matched by the controller. No matching, or persistency of excitation assumptions are needed.