  Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its
set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint
space. We show that the kernel of the maximal flat Radon transform $\tau:L^2(M)
\rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the
pullback map $L^2(\mathrm{ad}(M))\rightarrow L^2(M)$. In particular, we show
that the maximal flat Radon transform is injective if and only if $M$ coincides
with its adjoint space.
