  We show that K\"uchle fivefolds of type (c5) --- subvarieties of the
Grassmannian Gr(3,7) parameterizing 3-subspaces that are isotropic for a given
2-form and are annihilated by a given 4-form --- are birational to hyperplane
sections of the Lagrangian Grassmannian LGr(3,6) and describe in detail these
birational transformations. As an application, we show that the integral Chow
motive of a K\"uchle fivefold of type (c5) is of Lefschetz type. We also
discuss K\"uchle fourfolds of type (c5) --- hyperplane sections of the
corresponding K\"uchle fivefolds --- an interesting class of Fano fourfolds,
which is expected to be similar to the class of cubic fourfolds in many
aspects.
