We investigate a generalization of Brockett’s celebrated double bracket flow that is
closely related to matrix Riccati differential equations. Using known results on the classification of
transitive Lie group actions on homogeneous spaces, necessary and sufficient conditions for accessibility
of the generalized double bracket flow on Grassmann manifolds are derived. This leads to
sufficient Lie–algebraic conditions for controllability of the generalized double bracket flow. Accessibility
on the Lagrangian Grassmann manifold is studied as well, with applications to matrix Riccati
differential equations from optimal control.