  In this paper, we extend Gary R. Lawlor's original examples of
area-minimizing cones over nonoriented manifolds\cite{lawlor1991sufficient}.
From the point view of submanifolds themselves, by considering real nonoriented
Grassmannians, complex Grassmannians, quaternion Grassmannians and complex,
quaternion projective spaces and Cayley projective plane as the Hermitian
orthogonal projection operators uniformly, we prove that there exists a family
of opposite cones associated with them. These cones are shown area-minimizing
by Lawlor's Curvature Criterion, it also can be seen as direct proofs for these
cones being area-minimizing under the perspective of isolated orbits of adjoint
action and the perspective of symmetric spaces
(\cite{kerckhove1994isolated},\cite{hirohashi2000area},\cite{kanno2002area},\cite{ohno2015area}).
Additionally, for the left oriented real Grassmannian manifolds-the double
cover of the nonoriented cases, by embedding them in the exterior vector spaces
as unit simple vectors, we prove these cones are area-minimizing except one.
