  We study the varieties of reductions associated to the four Severi varieties,
the first example of which is the Fano threefold of index 2 and degree 5
studied by Mukai and others. We prove that they are smooth but very special
linear sections of Grassmann varieties, and rational Fano manifolds of
dimension $3a$ and index $a+1$, for $a=1, 2, 4, 8$. We study their maximal
linear spaces and prove that through the general point pass exactly three of
them, a result we relate to Cartan's triality principle. We also prove that
they are compactifications of affine spaces.
