The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. In view of memory efficient approximation, we review and extend known solution formulas and identify invariant subspaces for a possibly low-dimensional solution representation. Based on these theoretical findings, we propose a Galerkin projection onto a space related to a low-rank approximation of the algebraic Riccati equation. For the numerical implementation, we provide an alternative interpretation of the modified emphDavison-Maki method via the transformed flow of the differential Riccati equation, which enables us to rule out known stability issues of the method in combination with the proposed projection scheme. We present numerical experiments for large-scale autonomous differential Riccati equations and compare our approach with high-order splitting schemes.