K\"uchle fivefolds of type c5

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.