Broken symmetry solutions-solutions to the independent particle model that do not obey all symmetries required by the Hamiltonian-have attracted significant interest for capturing multireference properties with mean-field scaling. However, identification and optimization of broken-symmetry solutions is difficult owing to the non-linear nature of the self-consistent field (SCF) equations, particularly for solutions belonging to low-symmetry subgroups and where multiple broken symmetry solutions are sought. Linearization of SCF solution space results in the Lie algebra, which this work utilizes as a framework for elucidation of the set of solutions that exist at the desired symmetry. To demonstrate that searches constructed in the Lie algebra yield the set of broken symmetry solutions, a grid-based search of real-restricted, real-unrestricted, complex-restricted, paired-unrestricted, and real-general solutions of the (nearly ) H molecule is performed.