  We discuss the question of geometric formality for rationally elliptic
manifolds of dimension $6$ and $7$. We prove that a geometrically formal
six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a
symmetric space. We also show that a rationally hyperbolic six-dimensional
manifold with $b_2\leq 2$ and $b_3=0$ can not be geometrically formal. As it
follows from their real homotopy classification, the seven-dimensional
geometrically formal rationally elliptic manifolds have the real cohomology of
symmetric spaces as well.
